The point \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \) is on the terminal side of an angle. Find the six trig functions for the given angle. \( \sin \theta= \) \( \cos \theta= \) \( \tan \theta= \) What is the measure of the angle \( \theta \) in radians? \( \sec \theta= \) (Round to four decimal places) What is the measure of the angle in degrees?

To find the six trigonometric functions for the angle whose terminal side passes through the point \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\), we start by identifying the coordinates of the point as \( (x, y) \) where \( x = \frac{\sqrt{2}}{2} \) and \( y = \frac{\sqrt{2}}{2} \). ### Step 1: Calculate the radius \( r \) The radius \( r \) is calculated using the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the values of \( x \) and \( y \): \[ r = \sqrt{\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} = \sqrt{\frac{2}{4} + \frac{2}{4}} = \sqrt{\frac{2}{2}} = \sqrt{1} = 1 \] ### Step 2: Calculate the trigonometric functions Now we can find the six trigonometric functions: 1. **Sine**: \[ \sin \theta = \frac{y}{r} = \frac{\frac{\sqrt{2}}{2}}{1} = \frac{\sqrt{2}}{2} \] 2. **Cosine**: \[ \cos \theta = \frac{x}{r} = \frac{\frac{\sqrt{2}}{2}}{1} = \frac{\sqrt{2}}{2} \] 3. **Tangent**: \[ \tan \theta = \frac{y}{x} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 \] 4. **Cosecant** (the reciprocal of sine): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] 5. **Secant** (the reciprocal of cosine): \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] 6. **Cotangent** (the reciprocal of tangent): \[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{1} = 1 \] ### Summary of the trigonometric functions - \( \sin \theta = \frac{\sqrt{2}}{2} \) - \( \cos \theta = \frac{\sqrt{2}}{2} \) - \( \tan \theta = 1 \) - \( \csc \theta = \sqrt{2} \) - \( \sec \theta = \sqrt{2} \) - \( \cot \theta = 1 \) ### Step 3: Find the angle \( \theta \) The coordinates \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\) correspond to an angle of \( \frac{\pi}{4} \) radians (or 45 degrees) in the first quadrant. ### Step 4: Convert to degrees To convert radians to degrees: \[ \theta = \frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ \] ### Final Answers - \( \sin \theta = \frac{\sqrt{2}}{2} \) - \( \cos \theta = \frac{\sqrt{2}}{2} \) - \( \tan \theta = 1 \) - \( \sec \theta = \sqrt{2} \) (approximately \( 1.4142 \) when rounded to four decimal places) - Measure of the angle \( \theta \) in radians: \( \frac{\pi}{4} \) - Measure of the angle in degrees: \( 45^\circ \)