Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -510°

Verified step by step guidance

1. Step 1: Understand that the angle given is -510°, which is a negative angle. To find the trigonometric functions, first find a positive coterminal angle by adding 360° repeatedly until the angle is between 0° and 360°. Calculate: \(-510° + 360° = -150°\), then add 360° again: \(-150° + 360° = 210°\). So, the coterminal angle is \$210°$.
2. Step 2: Identify the reference angle for \$210°\(. Since \)210°\( is in the third quadrant (between 180° and 270°), the reference angle is \)210° - 180° = 30°$.
3. Step 3: Recall the exact values of the six trigonometric functions for the reference angle \$30°$: - \(\sin 30° = \frac{1}{2}\) - \(\cos 30° = \frac{\sqrt{3}}{2}\) - \(\tan 30° = \frac{1}{\sqrt{3}}\) - \(\csc 30° = 2\) - \(\sec 30° = \frac{2}{\sqrt{3}}\) - \(\cot 30° = \sqrt{3}\)
4. Step 4: Determine the signs of the trigonometric functions in the third quadrant. In the third quadrant, both sine and cosine are negative, but tangent is positive. Therefore: - \(\sin 210° = -\sin 30°\) - \(\cos 210° = -\cos 30°\) - \(\tan 210° = +\tan 30°\)
5. Step 5: Use the reciprocal identities to find the other three functions: - \(\csc 210° = \frac{1}{\sin 210°}\) - \(\sec 210° = \frac{1}{\cos 210°}\) - \(\cot 210° = \frac{1}{\tan 210°}\) Remember to rationalize denominators where applicable.

Key Concepts

Reference Angles and Coterminal Angles

To find trigonometric values for angles like -510°, first determine a coterminal angle between 0° and 360° by adding or subtracting 360°. Then, find the reference angle, which is the acute angle formed with the x-axis, to use known trigonometric values.

Signs of Trigonometric Functions in Different Quadrants

The sign of sine, cosine, and tangent depends on the quadrant where the angle lies. Knowing the quadrant helps assign the correct positive or negative sign to each function based on the ASTC (All Students Take Calculus) rule.

Rationalizing Denominators

When expressing exact trigonometric values, denominators containing radicals should be rationalized by multiplying numerator and denominator by a suitable radical. This simplifies the expression and adheres to standard mathematical conventions.