Questions: Find the exact values of the six trigonometric functions of the angle. -690° sin(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in cos(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in tan(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in cot(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers i csc(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers sec(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers

Find the exact values of the six trigonometric functions of the angle. -690° sin(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in cos(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in tan(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in cot(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers i csc(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers sec(-690°)= (Simplify your answer, including any radicals. Use integers or fractions for any numbers
Transcript text: Find the exact values of the six trigonometric functions of the angle. \[ \begin{array}{l} -690^{\circ} \\ \sin \left(-690^{\circ}\right)= \end{array} \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in \[ \cos \left(-690^{\circ}\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in \[ \tan \left(-690^{\circ}\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in \[ \cot \left(-690^{\circ}\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers i \[ \csc \left(-690^{\circ}\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers \[ \sec \left(-690^{\circ}\right)= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers
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Solution

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Solution Steps

Solution Approach

To find the exact values of the six trigonometric functions for the angle \(-690^\circ\), we first need to find the equivalent angle within the standard range of \([0^\circ, 360^\circ)\). This can be done by adding or subtracting multiples of \(360^\circ\) until the angle falls within this range. Once we have the equivalent angle, we can calculate the sine, cosine, and tangent values using Python's trigonometric functions. The other trigonometric functions (cotangent, cosecant, and secant) can be derived from these primary functions.

Step 1: Find the Equivalent Angle

To find the equivalent angle of \(-690^\circ\) within the range \([0^\circ, 360^\circ)\), we calculate: \[ -690^\circ \mod 360^\circ = 30^\circ \]

Step 2: Calculate the Sine Function

The sine of the angle is given by: \[ \sin(-690^\circ) = \sin(30^\circ) = 0.5 \]

Step 3: Calculate the Cosine Function

The cosine of the angle is given by: \[ \cos(-690^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \]

Step 4: Calculate the Tangent Function

The tangent of the angle is given by: \[ \tan(-690^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.5774 \]

Step 5: Calculate the Cotangent Function

The cotangent, being the reciprocal of the tangent, is: \[ \cot(-690^\circ) = \frac{1}{\tan(30^\circ)} = \sqrt{3} \approx 1.7321 \]

Step 6: Calculate the Cosecant Function

The cosecant, being the reciprocal of the sine, is: \[ \csc(-690^\circ) = \frac{1}{\sin(30^\circ)} = 2 \]

Step 7: Calculate the Secant Function

The secant, being the reciprocal of the cosine, is: \[ \sec(-690^\circ) = \frac{1}{\cos(30^\circ)} = \frac{2}{\sqrt{3}} \approx 1.1547 \]

Final Answer

The exact values of the six trigonometric functions for the angle \(-690^\circ\) are: \[ \sin(-690^\circ) = 0.5, \quad \cos(-690^\circ) = \frac{\sqrt{3}}{2}, \quad \tan(-690^\circ) = \frac{1}{\sqrt{3}}, \quad \cot(-690^\circ) = \sqrt{3}, \quad \csc(-690^\circ) = 2, \quad \sec(-690^\circ) = \frac{2}{\sqrt{3}} \] Thus, the final boxed answers are: \[ \boxed{\sin(-690^\circ) = 0.5} \] \[ \boxed{\cos(-690^\circ) = \frac{\sqrt{3}}{2}} \] \[ \boxed{\tan(-690^\circ) = \frac{1}{\sqrt{3}}} \]

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