Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of the sine, cosine, and tangent of the angle. $$-165^{\circ}$$

Step-by-Step Solution

Answer

Hence, the exact values of the sine, cosine, and tangent of the angle -165° are \(-\sin(15^{\circ}) = - \frac{\sqrt{6} - \sqrt{2}}{4}\), \(-\cos(15^{\circ}) = - \frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(15^{\circ}) = 2 - \sqrt{3}\) respectively.

Step 1: Converting to Positive Angle

Convert -165 degrees into a positive angle by adding 360 degrees. The equivalent positive angle is \( 360 - 165 = 195 \) degrees.

Step 2: Trigonometric Ratios for Sine

To find the sine of 195 degrees, you should know that sine values in the third quadrant (where 195 lies) are negative. For angle α in the third quadrant, the sine value is equal to the negative sine of its reference angle (180° - α). Hence, \(\sin(-165^{\circ}) = -\sin(15^{\circ})\).

Step 3: Trigonometric Ratios for Cosine

The cosine of angle α in the third quadrant is also negative. Hence, \(\cos(-165^{\circ}) = -\cos(15^{\circ})\).

Step 4: Trigonometric Ratios for Tangent

For the tangent, the ratio of sine/cosine gives the value of tangent. By substitution we get \( \tan(-165^{\circ}) = \frac{-\sin(15^{\circ})}{-\cos(15^{\circ})} = \tan(15^{\circ})\) as tangent is positive in the third quadrant.

Step 5: Calculating the Values

The final step is substituting the known values of the trigonometric ratios at 15 degrees. Here, we will use the known values: \(\sin(15^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4}\), \(\cos(15^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4}\), and \(\tan(15^{\circ}) = 2 - \sqrt{3}\).