91Ó°ÊÓ

To find the exact values for trigonometric functions, we begin by identifying the reference angle for each given angle. A reference angle is the smallest angle that a given angle forms with the x-axis. For any angle \(\theta\), the reference angle can be found as follows:- If \(\theta\) is in the first quadrant, the reference angle is \(\theta\) itself.- If \(\theta\) is in the second quadrant, the reference angle is \(180^\circ - \theta\) or \(\pi - \theta\) (in radians).- If \(\theta\) is in the third quadrant, the reference angle is \(\theta - 180^\circ \) or \(\theta - \pi\) (in radians).- If \(\theta\) is in the fourth quadrant, the reference angle is \(360^\circ - \theta\) or \(2\pi - \theta\) (in radians).

After determining the reference angle, we utilize known trigonometric values for reference angles \(30^\circ, 45^\circ, 60^\circ\) or their radian equivalents, which are \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\). For each angle, calculate all six trigonometric functions:- Sine \(\sin(\theta)\)- Cosine \(\cos(\theta)\)- Tangent \(\tan(\theta)\)- Cosecant \(\csc(\theta) = \frac{1}{\sin(\theta)}\)- Secant \(\sec(\theta) = \frac{1}{\cos(\theta)}\)- Cotangent \(\cot(\theta) = \frac{1}{\tan(\theta)}\)

The sign for each trigonometric function depends on the quadrant where the angle lies: - First Quadrant: All functions are positive. - Second Quadrant: Sine and cosecant are positive; others are negative. - Third Quadrant: Tangent and cotangent are positive; others are negative. - Fourth Quadrant: Cosine and secant are positive; others are negative.

Let's calculate each function for one example angle from the given list: \( 120^\circ \).1. Determine reference angle: \( 120^\circ \) is in the second quadrant, so reference angle is \( 180^\circ - 120^\circ = 60^\circ \).2. Use known values: - \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \) - \( \cos(60^\circ) = \frac{1}{2} \) - \( \tan(60^\circ) = \sqrt{3} \) - \( \csc(60^\circ) = \frac{2}{\sqrt{3}} \) - \( \sec(60^\circ) = 2 \) - \( \cot(60^\circ) = \frac{1}{\sqrt{3}} \)3. Apply sign for second quadrant: Sine and cosecant are positive; cosine, secant, tangent, and cotangent are negative. Therefore: - \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \) - \( \cos(120^\circ) = -\frac{1}{2} \) - \( \tan(120^\circ) = -\sqrt{3} \) - \( \csc(120^\circ) = \frac{2}{\sqrt{3}} \) - \( \sec(120^\circ) = -2 \) - \( \cot(120^\circ) = -\frac{1}{\sqrt{3}} \)

Repeat Steps 1 to 4 for each of the other angles given in the list, ensuring that you adjust for the quadrant and compute all six trigonometric functions accordingly.