### Still have math questions?

Q:

Find the distance between each pair of points. Round your answer to the nearest tenth. $$( - 0.6,7.1 ) , ( 6.8,5.6 )$$

A) $$3$$ B) $$7.6$$ C) $$14.1$$ D) $$11.1$$ 10) $$( 1.3,4.9 ) , ( - 1 , - 3.3 )$$

A) $$8.5$$ B) $$3.2$$ C) $$2.4$$ D) $$1.6$$

Q:

17. A ladder leaning against a wall makes an angle of $$75 ^ { \circ }$$ with the ground. If the foot of the ladder is $$6$$ feet from the base of the wall, what is the length of the ladder?

Q:

From a hot-air balloon, Grayson measures a $$29 ^ { \circ }$$ angle of depression to a landmark that's $$1004$$ feet away, measuring horizontally. What's the balloon's vertical distance above the ground? Round your answer to the nearest hundredth of a foot if necessary.

Q:

Use an algebraic equation to find the measures of the two angles described below. Begin by letting $$x$$ represent the degree measure of the angle's complement. The measure of the angle is $$82$$ degrees greater than its complement. What is the measure of the complement?

$$x = \square$$

Q:

Choose the letter of the equation for the graph. a. $$y = \sin ( x - \pi / 2 ) + 1$$

b. $$y = \cos ( x + \pi / 2 ) + 1$$

c. $$y = \cos ( x - \pi / 2 ) - 1$$

d. $$y = \cos ( x - \pi ) - 2$$

e. $$y = \sin ( x - \pi ) - 2$$

Q:

The diagram below represents a wheelchair ramp leading into a building. What is the height of the wheelchair ramp?

$$\square 80 cm$$

$$\square 79 cm$$

Q:

One angle of an isosceles triangle measures $$140 ^ { \circ }$$ . Which other angles could be in that isosceles triangle? Choose all that apply.

Q:

Write the complex number $$- 4 - 4 i$$ in trigonometric form, using degree measure for the argument.

$$- 4 - 4 i = \square ( \cos \square ^ { \circ } + i \sin \square ^ { \circ } )$$

(Type exact answers, using radicals as needed. Type any angle measures in degrees. Round to the nearest deg.

Q:

Sketch two periods of the graph of the function $$h ( x ) = 3 \sec ( \frac { \pi } { 4 } ( x + 1 ) )$$ . Identify the stretching factor, period, and asymptotes. Enter the exact answers.

Stretching factor $$=$$ Number Period:

$$P =$$

Enter the asymptotes of the function on the domain $$[ - P , P ]$$ . To enter $$\pi$$ , type Pi. The field below accepts a list of numbers or formulas separated by semicolons (e.g.

$$2 ; 4 ; 6$$ or $$x + 1 ; x - 1$$ ). The order of the list does not matter. Asymptotes:

$$x =$$

Select the correct graph of $$h ( x ) = 3 \sec ( \frac { \pi } { 4 } ( x + 1 ) )$$

Q:

Verify the following identity by rewriting the left side to look like the right side. Put the following steps in the correct order. $$\frac { \sin ( x + y ) } { \cos ( x - y ) } = \frac { \cot y + \cot x } { \cot x \cot y + 1 }$$

Q:

A function is of the form $$y = a \sin ( x ) + c$$ , where $$x$$ is in units of radians. If the valu a is $$4.75$$ and the value of $$c$$ is $$- 3$$ , what will the minimum of the function be? Input your answer as a decimal, rounded to two decimal places.

Q:

A function is of the form $$y = a \sin ( x ) + c$$ , where $$x$$ is in units of radians. If the value of a is $$37.50$$ and the value of $$c$$ is $$- 2$$ , what will the maximum of the function be? Input your answer as a decimal, rounded to two decimal places.

Q:

Question $$4$$ (1 point) A function is of the form $$y = \sin ( k x )$$ , where $$x$$ is in units of radians. If the period of the function is $$9$$ radians, what is the value of $$k$$ ? Input your answer as a decimal, rounded to two decimal places.

Q:

Question $$4$$ ( $$1$$ point) A function is of the form $$y = \sin ( k x )$$ , where $$x$$ is in units of radians. If the period of the function is $$9$$ radians, what is the value of $$k$$ ? Input your answer as a decimal, rounded to two decimal places.

Q:

Maria looks up at the top of a tree at an angle of elevation of $$75 ^ { \circ }$$ . She looks down at the bottom of the tree at an angle of depression of $$42 ^ { \circ }$$ . If she is standing $$2.5 m$$ away from the tree, how tall is the tree? Draw a diagram.

Q:

What are the domain and the range of $$y = \sin x ?$$

Find the domain of $$y = \sin x$$ . Choose the correct answer below.

A. The set of all real numbers B. The set of all real numbers except odd integer multiples of $$\frac { \pi } { 2 } ( 90 ^ { \circ } )$$

C. The set of all positive real numbers D. The set of all real numbers except integer multiples of $$\pi ( 180 ^ { \circ } )$$

Find the range of $$y = \sin x$$ . Choose the correct answer below.

A. $$y \leq 1$$ or $$y \geq - 1$$

B. $$- \infty < y < \infty$$

C. $$- 1 \leq y \leq 1$$

D. $$y \geq 1$$ or $$y \leq - 1$$

Q:

What are the domain and the range of $$y = \sin x ?$$

Find the domain of $$y = \sin x$$ . Choose the correct answer below.

A. The set of all real numbers

B. The set of all real numbers except odd integer multiples of $$\frac { \pi } { 2 } ( 90 ^ { \circ } )$$

C.. The set of all positive real numbers

D. The set of all real numbers except integer multiples of $$\pi ( 180 ^ { \circ } )$$

Find the range of $$y = \sin x$$ . Choose the correct answer below.

A. $$y \leq 1$$ or $$y \geq - 1$$

B. $$- \infty < y < \infty$$

(C. $$- 1 \leq y \leq 1$$

D. $$y \geq 1$$ or $$y \leq - 1$$

Q:

Solve. In triangle $$A B C$$ , the measure of angle $$C$$ is $$2$$ times the measure of angle B. Angle $$B$$ is $$24$$ less than double the angle A. Find the measures of each angle in degrees.

(A) $$A = 72 , B = 36 , C = 72$$

(B) $$A = 36 , B = 48 , C = 96$$

(C) $$A = 36 , C = 96 , B = 48$$

(D) $$A = 56 , B = 100 , C = 24$$

Q:

1.Casey sights the top of an $$84$$ -foot tall lighthouse at an angle of elevation of $$58 ^ { \circ }$$ . If Casey is $$6$$ feet tall, how far is he standing from the base of the lighthouse?

2.The angle of elevation from a kicker's foot on the football field to the top of the goal post bars is $$17 ^ { \circ }$$ . If he is standing $$131$$ feet from the base of the goal post, how tall is the goal post?

3. Leah's mom is standing at the bottom of the slide at the playground, waiting for Leah to slide down. If the angle of elevation from the bottom of the slide to the top is $$46 ^ { \circ }$$ , and the slide has a vertical height of $$9$$ feet, find the length of the slide.

4. A dog is standing $$5$$ feet from the base of a tree, looking up at a cat that has climbed $$16$$ feet up the tree. What is the angle of elevation from the point the dog is standing on the ground to the cat?

Q:

Base your answer to the following question on In the accompanying diagram, $$\Delta F U N$$ is a right triangle, $$\overline { U R }$$ is the altitude to hypotenuse $$\overline { F N } , U R = 12$$ , and the lengths of $$\overline { F R }$$ and $$\overline { R N }$$ are in the ratio $$1 : 9$$ .

Q:

When proving that a triangle is a right triangle using coordinate geometry methods, you must: A. show that the slopes of two sides are negative reciprocals creating perpendicular lines and right angles. B. show that the lengths of the sides satisfy the Pythagorean Theorem, thus creating a right triangle. C. Both choices A and B may be used D. Neither A or B may be used

Q:

The point $$P ( 4 , - 3 )$$ lies on the terminal arm of an angle $$q$$ in standard position. Determine the measure of $$q$$ to the nearest degree. Select one:

a. $$- 143 ^ { \circ }$$

b. $$127 ^ { \circ }$$

c. $$323 ^ { \circ }$$

d. $$233 ^ { \circ }$$

Q:

A triangle has angle measurements of $$90 ^ { \circ } , 27 ^ { \circ } ,$$ and $$63 ^ { \circ } .$$ What type of triangle is this?

Acute

Right

Obtuse

Q:

8. In $$\triangle Q R S , q = 1.7 m , r = 4.3 m$$ , and $$s = 5.6 m$$ . Solve $$\triangle Q R S$$ by determining the measure of each angle to the nearest degree. Show your work and draw $$\triangle Q R S$$ . ( $$6$$ marks)

Q:

8. In $$\triangle Q R S , q = 1.7 m , r = 4.3 m$$ , and $$s = 5.6 m$$ . Solve $$\triangle Q R S$$ by determining the measure of each angle to the nearest degree. Show your work and draw $$\triangle Q R S$$ . ( $$6$$ marks)

Q:

If the tan of angle $$x$$ is $$\frac { 22 } { 5 }$$ and the triangle was dilated to be two times as big as the original, what would be the value of the tan of $$x$$ for the dilated triangle? Clue: Use the slash symbol ( $$I$$ ) to represent the fraction bar, and enter the fraction with no spaces. Blank 1:

Q:

Isabell is standing at the top of a waterslide that leads to a pool below. The angle of depression from top of the slide to the pool is $$31.66 ^ { \circ }$$ , and the slide is $$46$$ feet long. How far is the top of the slide from the ground? Round your answer to the nearest foot. a $$22$$ feet b $$24$$ feet a feet a feet

Q:

Find the cosine ratio of angle $$\Theta .$$ Clue: Use the slash symbol ( $$/ )$$ to represent the fraction bar, and enter the fraction with no spaces.

Q:

If you use the following information to construct $$\triangle ABC$$ , the location of side $$b$$ should be ___

$$\angle A = 90 ^ { \circ }$$

$$\angle C = 60 ^ { \circ }$$

$$b = 4$$ units

Q:

Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is ( ) than the length of the third side.

Q:

A person on the top of a tall building looks through his binoculars at his friend that is $$300$$ ft away from the building on the ground. If the angle of depression from the person on the building is $$30 ^ { \circ }$$ , how tall is the building? Round your answer to the nearest foot.

$$158 ft$$

$$148 ft$$

$$152 ft$$

Q:

A six foot man is standing $$80$$ feet away from a building. He looks up at a $$30 ^ { \circ }$$ angle to see the top of the building. The building is $$52$$ feet tall. Round to the nearest foot when determining if the answer is true or false. True

Q:

If a person stands $$40 ft$$ from a tree and the angle of elevation made from their eyes ( $$5 ft$$ above the ground) to the top of the tree is $$47 ^ { \circ }$$ , how tall is the tree? Round your answer to the nearest foot.

$$52 ft$$

$$58 ft$$

$$36 ft$$

Q:

The population of predators and prey in a closed ecological system tends to vary periodically over time. In a certain system, the population of owls $$O$$ can be represented by$$O = 120 + 15 \sin ( \frac { \pi } { 15 } t )$$ where $$t$$ is the time in years since January $$1,2001 .$$ In that same system, the population of mice $$M$$ can be represented by$$M = 500 + 200 \sin ( \frac { \pi } { 15 } t + \frac { \pi } { 30 } ) .$$ What is the maximum number of owls and how many years does it take to reach this population for the first time?

$$120$$ owls, $$15$$ years

$$120$$ owls, $$7.5$$ years

$$135$$ owls, $$15$$ years

$$135$$ owls, $$7.5$$ years

Q:

Find the exact value of each expression. Write angle measures in degrees.

6. $$\cos ( - 120 ^ { \circ } )$$

7. $$\sec ( - \frac { 7 \pi } { 6 } )$$

8. $$\operatorname { Sin } ^ { - 1 } ( - \frac { \sqrt { 3 } } { 2 } )$$

9. $$\operatorname { Arctan } - 1$$

Q:

Evaluate the integral: $$\int \frac { 30 x ^ { 2 } } { \sqrt { 36 - x ^ { 2 } } } d x$$

(A) Which trig substitution is correct for this integral?

$$x = 6 \sin ( \theta )$$

$$x = 36 \sec ( \theta )$$

$$x = 6 \sec ( \theta )$$

$$x = 6 \tan ( \theta )$$

(B) Which integral do you obtain after substituting for $$x$$ and $$\operatorname { simplifying? }$$

Note: to enter $$\theta ,$$ type the word theta.

$$\int x = 36 \tan ( \theta )$$

(C) What is the value of the above integral in terms of $$\theta ?$$

(D) What is the value of the original integral in terms of $$x ?$$

$$x = 36 \sin ( \theta )$$

Q:

Triangle DEF contains right angle E. If angle D measures $$50 ^ { \circ }$$ and its opposite side measures $$7.6$$ units, what is the measure of side DE? Round your answer to the nearest hundredth.

a $$4.89$$ units

b $$5.34$$ units

c $$6.38$$ units

$$9.06$$ units

Q:

Which set of angle measures could be the interior angles of a triangle?

A. $$60 ^ { \circ } , 90 ^ { \circ } , 120 ^ { \circ }$$

B. $$35 ^ { \circ } , 60 ^ { \circ } , 75 ^ { \circ }$$

C. $$25 ^ { \circ } , 30 ^ { \circ } , 35 ^ { \circ }$$

D. $$45 ^ { \circ } , 60 ^ { \circ } , 75 ^ { \circ }$$

Q:

27-30 Find the first four distinct Taylor polynomials about

$$x = x _ { 0 }$$ , and use a graphing utility to graph the given function and the Taylor polynomials on the same screen. 27. $$f ( x ) = e ^ { - 2 x } ; x _ { 0 } = 0$$ 28. $$f ( x ) = \sin x ; x _ { 0 } = \pi / 2$$

29. $$f ( x ) = \cos x ; x _ { 0 } = \pi$$ 30. $$\ln ( x + 1 ) ; x _ { 0 } = 0$$

Q:

21. Practice similar Help me with this Let $$s ( t )$$ denote the position of a particle at time $$t ,$$ and let $$v$$ and $$a$$ be the velocity and acceleration respectively. The particle is moving according to the data $$( t ) = \cos ( t ) + \sin ( t ) , s ( 0 ) = 8 , v ( 0 ) = 5 .$$

Find a function describing the position of the particle.

$$s ( t ) =$$

Q:

Use the ALEKS calculator to evaluate each expression. Give your answers in radians. Round them to the nearest hundredth. If applicable, click or "Undefined."

$$\tan ^ { - 1 } ( - 2.42 ) = \square$$

$$\cos ^ { - 1 } ( 1.86 ) = \square$$

$$\sin ^ { - 1 } ( 0.42 ) = \square$$

Q:

In triangle $$A B C$$ , the length of side $$A B$$ is $$10$$ inches and the length of side $$B C$$ is $$17$$ inches. Which of the following could be the length of side $$A C$$ ?

$$32$$ inches   $$5$$ inches   $$29$$ inches   $$16$$ inches

Q:

If the longest side of the triangle was $$10$$ . Then the two shorter sides must have a sum that is ...? less than $$10$$  any length  greater than $$10$$  equal to 10

Q:

27. A pilot flies $$90$$ miles from Memphis, TN to Tupelo, MS, then $$122$$ miles from there to Hunstville, AL, and then flies back to Memphis. How long was the flight from Hunstville to Memphis? Memphis, TN

Q:

26. Two cars start moving from the same location. They head straight, but in different directions. The angle between where they are heading is $$43 ^ { \circ }$$ . The first car travels $$20$$ miles and the second car travels $$37$$ miles. How far apart are the two cars? Round your answer to the nearest tenth.

Q:

21. Dakota's eyes are $$6 ft$$ off the ground, he is looking up to the top of the Empire State building at an angle of $$80 ^ { \circ }$$ . If he is standing $$255.3 ft$$ away from the building, how tall is it? (Don't forget to add in Dakota's height.) a.) Draw a picture. b.) How high is the building?

Q:

3) A weather balloon is rising vertically at the rate of $$5 ft / s$$ . An observer is standing on the ground $$400 ft$$ horizontally from the point where the balloon was released. a) At what rate is the distance between the observer and the balloon changing when the balloon is $$300 ft$$ high? b) At what rate is the angle of elevation of the observer's line of sight to the balloon changing when the balloon is $$300$$ feet high? 4) Use Newton's Method to find all roots of the following equation. Answers should be correct to three decimal places.

Q:

Question $$6$$ An airplane is traveling due west with a speed of $$300$$ miles per hour. The wind blows at $$60$$ miles per hour due south. 1. What is the resultant speed of the airplane's flight? Round your answer to the nearest whole number. 2. What is the resultant direction of the airplane's flight? Round your answer to the nearest degree. Remember direction must be answered as an angle measured from due east (positive $$x$$ -axis)

Q:

From home, Brett walked northeast $$8$$ miles to an old tree stump. He walked southeast an additional $$18$$ miles to the lake. Which one of the choices below is a possible distance between Brett's house and the lake?

Q:

Triangle $$ABC$$ has coordinates $$A ( - 6,2 )$$ ,

$$B ( - 3,6 )$$ , and $$C ( 5,0 )$$ . Find the perimeter of the triangle. Round your answer to the hundredths place (2 decimal places).

Q:

Given a triangle $$A B C$$ at points $$A = ( - 1 , - 4 ) B = ( 2,6 ) C = ( 6,2 )$$ , and if the triangle is dilated with a scale factor of $$4$$ , find the new point $$C$$ '. Select one: a. $$( 6,8 )$$

b. $$( 24,8 )$$

c. $$( 4,1 )$$

d. $$( 6,4 )$$

Q:

Given a triangle $$A B C$$ at points $$A = ( 3,5 ) B = ( 1,6 ) C = ( 7,1 )$$ , and a first transformation of right $$1$$ and up $$2$$ , and a second transformation of left $$6$$ and down $$4$$ , what would be the location of the final point

$$B ^ { \prime \prime }$$ ? Select one: a. $$( 0,4 )$$

b. $$( 1,1 )$$

c. $$( - 4,4 )$$

d. $$( - 4,1 )$$

Q:

Given a triangle $$A B C$$ at points $$A = ( - 2,2 ) B = ( 2,5 ) C = ( 2,0 )$$ , and a first transformation of right $$4$$ and up $$3$$ , and a second transformation of left $$2$$ and down $$5$$ , what would be the location of the final point $$B ^ { \prime \prime }$$ ? Select one: a. $$( 2 , - 1 )$$

b. $$( 4,8 )$$

c. $$( 4,3 )$$

d. $$( 6,3 )$$

Q:

In the figure below, $$\triangle R S T$$ could be translated four units to the left to form $$\triangle X Y Z$$ .

Kip claims that when a triangle is translated, it becomes impossible to determine the measures of the new triangle. His teacher tells him that it is possible to determine the measures of the sides of $$\triangle X Y Z$$ . Complete the table below to determine the measures of the sides of $$\triangle X Y Z$$ .

Q:

Given the triangle $$A B C$$ at points $$A = ( 1 , - 4 ) B = ( 4 , - 5 ) C = ( 6 , - 3 )$$ , and if the triangle is first reflected over the $$y$$ axis, and then over the $$x$$ axis, find the new point $$A ^ { \prime \prime } .$$

Select one: a. $$( - 1,4 )$$

b. $$( 3,2 )$$

c. $$( - 1 , - 4 )$$

d. $$( 1,4 )$$

Q:

Given a triangle $$A B C$$ at points $$A = ( - 3,4 ) B = ( 4,8 ) C = ( 6,3 ) ,$$ and if the triangle is dilated with a scale factor of $$1.5$$ , find the new point $$B ^ { \prime }$$ . Select one: a. $$( 2,4 )$$

b. $$( 5.5,9.5 )$$

c. $$( 4,8 )$$

d. $$( 6,12 )$$

Q:

To do a translation of a polygon, it is easier to move one point at a time, rather than focus on the whole polygon. Select one: True False

Q:

Given the triangle $$A B C$$ at points $$A = ( 1,6 ) B = ( - 3,5 ) C = ( 7,1 )$$ , and if the triangle is first reflected over the $$y$$ axis, and then over the $$x$$ axis, find the new point $$A ^ { \prime \prime }$$ . Select one: a. $$( 1 , - 6 )$$

b. $$( - 1 , - 6 )$$

c. $$( 3,5 )$$

d. $$( 1,6 )$$

Q:

Label the triangle below with the angle of elevation and angle of depression. And explain how to use the angle of depression ( $$4$$ marks)

Q:

A triangle has vertices $$T ( 8 , - 4 ) , U ( 0,8 ) ,$$ and $$V ( - 6 , - 8 )$$ . The midpoints of

$$TU$$ and $$TV$$ are $$W$$ and $$Z$$ , respectively. a) Graph the triangle. Determine the coordinates of $$W$$ and $$Z$$ . b) Calculate the lengths of $$WZ$$ and $$UV$$ . What do you notice?

Q:

A right triangle has vertices $$P ( 5 , - 5 ) , Q ( 3 , - 10 ) ,$$ and $$R ( - 10,1 ) .$$

a) Draw the triangle on a grid. Determine the coordinates of the midpoint $$S$$ of the hypotenuse. b) Calculate the lengths of PS, QS, and RS. What do you notice? Explain.

Q:

In right triangle $$P R Q , \overline { P Q }$$ is the hypotenuse and $$m \angle P = 58$$ . What is the shortest side of $$\triangle P R Q ?$$

Q:

Convert the degrees measure $$45 ^ { \circ }$$ to radian measure. Please enter a fraction. Answer:

$$\pi$$

Q:

Find the length of AB to the nearest tenth of a metre. Hint: using $$\triangle A C D$$ , find the length of $$A C$$ (tan) using $$\triangle B C D$$ , find the length of $$B C$$ (tan) subtract $$A C - B C$$ to find the length of $$A B$$

Q:

Two secants $$C B$$ and $$E D$$ of a circle meet in point $$A$$ outside the circle. DF is a tangent to the circle at D. $$\angle B D F = \angle F A D$$

$$= 30 ^ { \circ } .$$ Which of the following statements is necessarily true? 1] $$G$$ is the centre of the circle 2] $$B E$$ is perpendicular to $$C D$$ 3] $$\angle C E B$$ measures $$45 ^ { \circ }$$

4] $$l ( A B ) + I ( B E ) = I ( A D ) + l ( D C )$$

Q:

In circle $$A$$ below, chord $$BC$$ and diameter $$DAE$$ intersect at $$F$$ . If $$m \overline { C D } = 46 ^ { \circ }$$ and $$m \overline { D B } = 102 ^ { \circ }$$ , what is $$m < CFE$$ ?

Q:

What is the value of $$x$$ in the figure below? In this diagram, $$\triangle A B D \sim \triangle C A D$$ .

A. $$45$$ B. $$\sqrt { 70 }$$

C. $$\frac { 9 } { 14 }$$

D. $$\sqrt { 126 }$$

E. $$\sqrt { 45 }$$

F. $$14$$

Q:

Find the exact value of the side $$s$$ in the triangle below. The angle $$t = 30$$ degrees. $$t$$ is noted in red below.

Q:

In $$\triangle CDE$$ , the measure of $$\angle E = 90 ^ { \circ } , DC = 97 , CE = 72$$ , and $$ED = 65 .$$ What ratio represents the sine of $$\angle C$$ ?

Q:

Questlon $$3$$ of $$10$$ A certain triangle has a $$30 ^ { \circ }$$ angle and a $$60 ^ { \circ }$$ angle. Which must be a true statement about the triangle?

A. Two sides of the triangle have the same length.

B. The longest side is $$\sqrt { 3 }$$ times as long as the shortest side.

C. The second-longest side is $$\sqrt { 3 }$$ times as long as the shortest side.

D. The longest side is twice as long as the second-longest side.

Q:

2. Find the exact values for the $$6$$ trigonometric ratios for angle A given the side lengths:

$$a = 8$$ and $$c = 17$$ . Remember to rationalize the denominators when necessary. (3 pts total)

$$\operatorname { Sin } A = \operatorname { Csc } A =$$

$$\operatorname { Cos } A = \operatorname { Sec } A =$$

$$\operatorname { Tan } A = \operatorname { Cot } A =$$

Q:

Triangle XYZ was dilated by a scale factor of $$2$$ to create triangle $$A C B$$ and $$\sin \angle X = \frac { 5 } { 5.59 }$$ . Part A: Use complete sentences to explain the special relationship between the trigonometric ratios of triangles $$X Y Z$$ and ACB. You must show all work and calculations to receive full credit. ( $$5$$ points)

Part B: Explain how to find the measures of segments CB and AB. You must show all work and calculations to receive full credit. (5 points)

Q:

7. Which reason proves that $$\triangle F H J \sim \Delta I G J$$ ?

(A) AA Similarity Postulate (B) SAS Similarity Theorem

(C) SSS Similarity Theorem (D) Third Angles Theorem

8. Which reason proves that $$\triangle ABC \sim \triangle GHI$$ ? Explain your answer.

Q:

11. Which equation can be used to find the value of $$x$$ ?

(A) $$\frac { x } { 6 } = \frac { 2 } { 5 }$$

(B) $$\frac { x } { 6 } = \frac { 5 } { 2 }$$

(C) $$\frac { x } { 5 } = \frac { 6 } { 2 }$$

(D) $$\frac { x } { 5 } = \frac { 2 } { 6 }$$

12. If $$\overline { N P }$$ bisects $$\angle M N O ,$$ what is the measure of MP?

Q:

The measures, in degrees, of the three angles of a triangle are given by $$2 x + 1$$ , $$3 x - 3$$ , and $$9 x$$ . What is the measure of the smallest angle?

(A) $$13 ^ { \circ }$$ (C) $$36 ^ { \circ }$$

(B) $$27 ^ { \circ }$$ (D) $$117 ^ { \circ }$$

Q:

Evaluating Trigonometric Functions of $$30 ^ { \circ } , 45 ^ { \circ }$$ , and $$60 ^ { \circ }$$ In Exercises $$23 - 28$$ , construct an appropriate triangle to find the missing values. $$( 0 ^ { \circ } \leq \theta \leq 90 ^ { \circ } , 0 \leq \theta \leq \pi / 2 )$$

Q:

Evaluating Sine, Cosine, and Tangent In Exercises 13-22, evaluate (if possible) the sine, cosine, and tangent at the real number. 13. $$t = \frac { \pi } { 4 }$$

15. $$t = - \frac { \pi } { 6 }$$

Q:

EXAMPLE $$1$$ Evaluating Trigonometric Functions See LarsonPrecalculus.com for an interactive version of this type of example. Evaluate the six trigonometric functions at each real number. a. $$t = \frac { \pi } { 6 }$$ b. $$t = \frac { 5 \pi } { 4 }$$ c. $$t = \pi$$ d. $$t = - \frac { \pi } { 3 }$$

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$$5.4$$ Evaluating Trigonometric Functions In Exercises $$47 - 50$$ , find the exact values of the sine, cosine, and tangent of the angle. 47. $$75 ^ { \circ } = 120 ^ { \circ } - 45 ^ { \circ }$$

49. $$\frac { 25 \pi } { 12 } = \frac { 11 \pi } { 6 } + \frac { \pi } { 4 }$$

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25. Angle of Elevation An engineer designs a $$75$$ -foot cellular telephone tower. Find the angle of elevation to the top of the tower at a point on level ground $$50$$ feet from its base.

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EXAMPLE $$8$$ Solving a Right Triangle A surveyor stands $$115$$ feet from the base of the Washington Monument, as shown in Figure $$4.24$$ . The surveyor measures the angle of elevation to the top of the monument to be $$78.3 ^ { \circ }$$ . How tall is the Washington Monument?

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EXAMPLE $$10$$ Solving a Right Triangle Find the length $$c$$ and the height $$b$$ of the skateboard ramp below.

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93. Angle of Elevation The height of a radio transmission tower is $$70$$ meters, and it casts a shadow of length $$30$$ meters. Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. Then find the angle of elevation.

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5. Surveying A surveyor in a helicopter is determining the width of an island, as shown in the figure. (a) What is the shortest distance $$d$$ the helicopter must travel to land on the island? (b) What is the horizontal distance $$x$$ the helicopter must travel before it is directly over the nearer end of the island? (c) Find the width $$w$$ of the island. Explain how you found your answer.

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Evaluating Trigonometric Functions In Exercises

$$1 1 - 1 4$$ , find the exact values of the six trigonometric functions of the angle $$\theta$$ for each of the two triangles. Explain why the function values are the same.

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Simplifying a Trigonometric Expression In Exercises $$33 - 40$$ , use the fundamental identities to simplify the expression. (There is more than one correct form of each answer.) 33. $$\tan \theta \csc \theta$$

35. $$\sin \phi ( \csc \phi - \sin \phi )$$

37. $$\sin \beta \tan \beta + \cos \beta$$

39. $$\frac { 1 - \sin ^ { 2 } x } { \csc ^ { 2 } x - 1 }$$

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frigonometric Substitution In Exercises $$57$$ and

$$58$$ , use the trigonometric substitution to write the algebraic equation as a trigonometric equation of $$\theta$$ , where $$- \pi / 2 < \theta < \pi / 2$$ . Then find $$\sin \theta$$ and $$\cos \theta$$ . 57. $$\sqrt { 2 } = \sqrt { 4 - x ^ { 2 } } , x = 2 \sin \theta$$

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Multiplying Trigonometric Expressions In Exercises $$4 1$$ and $$4 2$$ , perform the multiplication and use the fundamental identities to simplify. (There is more than one correct form of each answer.) 41. $$( \sin x + \cos x ) ^ { 2 }$$

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Trigonometric Substitution In Exercises $$53 - 56$$ , use the trigonometric substitution to write the algebraic expression as a trigonometric function of $$\theta$$ , where

$$0 < \theta < \pi / 2$$ . 53. $$\sqrt { 9 - x ^ { 2 } } , x = 3 \cos \theta$$

55. $$\sqrt { x ^ { 2 } - 4 } , x = 2 \sec \theta$$

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The angle bisectors of $$\triangle X Y Z$$ are $$\overline { X G } , \overline { Y G }$$ , and $$\overline { Z G }$$ . They meet at a single point $$G$$ . (In other words, $$G$$ is the incenter of $$\triangle X Y Z$$ .) Suppose $$D G = 12 , X G = 17 , m \angle D X F = 98 ^ { \circ }$$ , and $$m \angle F Z G = 14 ^ { \circ }$$ . Find the following measures. Note that the figure is not drawn to scale. $$m \angle F Z E = \square$$

$$m \angle E Y G = \square$$

$$E G = \square$$

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Y= 3√x

D 5. Find the inverse. $$y = \sqrt { 3 x }$$

$$y = ( x ^ { 3 } )$$

$$x = y ^ { 3 }$$

$$y = \sqrt { x ^ { 3 } }$$

$$y = 3 x$$

$$\overline { y } =$$

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Triangle $$A B C$$ is plotted in the coordinate plane. The triande is rotated $$\ 0$$ degrees clockwise and left two. What are the coordirafes for CF?

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Find the length of the hypotenuse.

$$5$$

$$18$$

$$18$$

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19. Tell if the triangles are congruent by SSS Name the pair of congruent triangles. (Don't forget about shared sides or vertical angles.) The triangles are congruent by $$\Delta M N L \cong \Delta$$

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If $$m \angle 1 = x$$ and $$90 ^ { \circ } < x < 180 ^ { \circ } ,$$ then $$\angle 1$$ is $$a ( n ) :$$

straight angle acute angle obtuse angle right angle

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The team from "How Ridiculous" are trying to make a world record basketball basket. The thrower is on a cliff $$250 f t$$ high and the angle of depression is $$15$$ degrees. How far did the basketball travel?(Use the graphic to help) Rounded to the nearest foot. "Graphic not drawn to scale * $$65 ft$$

$$259 ft$$

$$242 ft$$

$$966 ft$$

Not listed

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Express 13∠83 in rectangular notation.

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Give exact answer and an approximation to three decimal places.

In the right triangle, find the length of the side not given. Give an exact answer and an approximation to three decimal places. $$a = 12 , b = 12$$The exact value of $$c$$ is

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Coordinates for a polygon are $$( 1,4 ) , ( 6,4 )$$ , and $$( 6,1 )$$ . What type of polygon is formed by these points? (hint: graph the figure and then apply properties) Right Triangle Trapezoid Equilateral Triangle Isosceles Triangle

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Find all missing side and angle of the triangle. If there are two solutions,given both if there is no triangle,state this and indicate how you know A=40 °, a=8, b=10

2. Find all missing sides and angles of the triangle. If there are two solutions, give both. If there is no triangle, state this and indicate how you know.

$$A = 40 ^ { \circ } , a = 8 , b = 10$$

(5 points)