Question

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -390°

Accepted Answer

Step 1: Understand that the angle given is -390°, which is a negative angle. To find the trigonometric functions, first convert this angle to a positive coterminal angle by adding 360° repeatedly until the angle is between 0° and 360°. So, calculate $$-390° + 360° = -30°. $$Since -30° is still negative, add 360° again: $$-30° + 360° = 330°. $$Thus, the coterminal positive angle is 330°. Step 2: Identify the reference angle for 330°. The reference angle is the acute angle formed with the x-axis. Since 330° is in the fourth quadrant, the reference angle is $$360° - 330° = 30°. $$Step 3: Recall the exact trigonometric values for 30°: $$\sin 30° = \frac{1}{2}$$, $$\cos 30° = \frac{\sqrt{3}}{2}$$, and $$	an 30° = \frac{1}{\sqrt{3}}. $$Use these as the basis for the values at 330°, adjusting signs according to the quadrant. Step 4: Determine the signs of the trigonometric functions in the fourth quadrant (where 330° lies). In the fourth quadrant, cosine is positive, sine is negative, and tangent is negative. Apply these signs to the reference angle values to find $$\sin 330°$$, $$\cos 330°$$, and $$	an 330°. $$Step 5: Use the primary sine, cosine, and tangent values to find the remaining three trigonometric functions: cosecant ($$\csc 	heta = \frac{1}{\sin 	heta}$$), secant ($$\sec 	heta = \frac{1}{\cos 	heta}$$), and cotangent ($$\cot 	heta = \frac{1}{	an 	heta}). $$Remember to rationalize denominators where necessary.

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -390°

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Concetti chiave

Reference Angles and Coterminal Angles

Signs of Trigonometric Functions in Different Quadrants

Exact Values and Rationalizing Denominators