Find the remaining five trigonometric functions of satisfying the conditions. (Hint: Draw a triangle
in the appropriate quadrant. Pay attention to signs!
14.

Ask by Chen Rodriquez. in the United States

Mar 20,2025

Question

Find the remaining five trigonometric functions of satisfying the conditions. (Hint: Draw a triangle
in the appropriate quadrant. Pay attention to signs!
14.

Ask by Chen Rodriquez. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

To find the remaining five trigonometric functions of $$ \theta $$ given that $$ \sec \theta = \frac{6}{5} $$ and $$ \tan \theta \leq 0 $$, we can follow these steps:

1. **Identify the quadrant**: Since $$ \sec \theta $$ is positive and $$ \tan \theta $$ is negative, $$ \theta $$ must be in the fourth quadrant (where cosine is positive and sine is negative).

2. **Find cosine**: The secant function is the reciprocal of the cosine function:
   $$
   \sec \theta = \frac{1}{\cos \theta} \implies \cos \theta = \frac{5}{6}
   $$

3. **Use the Pythagorean identity**: We can find sine using the identity:
   $$
   \sin^2 \theta + \cos^2 \theta = 1
   $$
   Substituting for $$ \cos \theta $$:
   $$
   \sin^2 \theta + \left(\frac{5}{6}\right)^2 = 1
   $$
   $$
   \sin^2 \theta + \frac{25}{36} = 1
   $$
   $$
   \sin^2 \theta = 1 - \frac{25}{36} = \frac{36}{36} - \frac{25}{36} = \frac{11}{36}
   $$
   $$
   \sin \theta = -\sqrt{\frac{11}{36}} = -\frac{\sqrt{11}}{6}
   $$
   (We take the negative root because sine is negative in the fourth quadrant.)

4. **Find tangent**: The tangent function is given by:
   $$
   \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{11}}{6}}{\frac{5}{6}} = -\frac{\sqrt{11}}{5}
   $$

5. **Find cosecant**: The cosecant function is the reciprocal of sine:
   $$
   \csc \theta = \frac{1}{\sin \theta} = -\frac{6}{\sqrt{11}}
   $$

6. **Find cotangent**: The cotangent function is the reciprocal of tangent:
   $$
   \cot \theta = \frac{1}{\tan \theta} = -\frac{5}{\sqrt{11}}
   $$

Now we have all six trigonometric functions:
- $$ \sin \theta = -\frac{\sqrt{11}}{6} $$
- $$ \cos \theta = \frac{5}{6} $$
- $$ \tan \theta = -\frac{\sqrt{11}}{5} $$
- $$ \csc \theta = -\frac{6}{\sqrt{11}} $$
- $$ \sec \theta = \frac{6}{5} $$
- $$ \cot \theta = -\frac{5}{\sqrt{11}} $$

Thus, the remaining five trigonometric functions of $$ \theta $$ are:
- $$ \sin \theta = -\frac{\sqrt{11}}{6} $$
- $$ \tan \theta = -\frac{\sqrt{11}}{5} $$
- $$ \csc \theta = -\frac{6}{\sqrt{11}} $$
- $$ \cot \theta = -\frac{5}{\sqrt{11}} $$ 
- $$ \sec \theta = \frac{6}{5} $$ (given)
The remaining five trigonometric functions of $$ \theta $$ are: - $$ \sin \theta = -\frac{\sqrt{11}}{6} $$ - $$ \tan \theta = -\frac{\sqrt{11}}{5} $$ - $$ \csc \theta = -\frac{6}{\sqrt{11}} $$ - $$ \cot \theta = -\frac{5}{\sqrt{11}} $$ - $$ \sec \theta = \frac{6}{5} $$

Find the remaining five trigonometric functions of satisfying the conditions. (Hint: Draw a triangle
in the appropriate quadrant. Pay attention to signs!
14.

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Find the remaining five trigonometric functions of satisfying the conditions. (Hint: Draw a triangle in the appropriate quadrant. Pay attention to signs! 14.

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Find the remaining five trigonometric functions of satisfying the conditions. (Hint: Draw a triangle
in the appropriate quadrant. Pay attention to signs!
14.