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

Ask by Campos Turnbull. 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!
914. QII III

Ask by Campos Turnbull. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

To find the remaining five trigonometric functions of $$ \theta $$ given the conditions, we will analyze each case separately. ### Case 1: $$ \sec \theta = \frac{6}{2} $$, $$ \tan \theta < 0 $$, $$ \cos \theta < 0 $$ (Quadrant II) 1. **Calculate $$ \sec \theta $$**: $$ \sec \theta = \frac{6}{2} = 3 $$ Since $$ \sec \theta = \frac{1}{\cos \theta} $$, we can find $$ \cos \theta $$: $$ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{3} $$ However, since $$ \cos \theta < 0 $$ in Quadrant II, we have: $$ \cos \theta = -\frac{1}{3} $$ 2. **Find $$ \sin \theta $$** using the Pythagorean identity: $$ \sin^2 \theta + \cos^2 \theta = 1 $$ $$ \sin^2 \theta + \left(-\frac{1}{3}\right)^2 = 1 $$ $$ \sin^2 \theta + \frac{1}{9} = 1 $$ $$ \sin^2 \theta = 1 - \frac{1}{9} = \frac{8}{9} $$ $$ \sin \theta = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3} $$ Since $$ \sin \theta > 0 $$ in Quadrant II, we have: $$ \sin \theta = \frac{2\sqrt{2}}{3} $$ 3. **Find $$ \tan \theta $$**: $$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{2}}{3}}{-\frac{1}{3}} = -2\sqrt{2} $$ 4. **Find $$ \csc \theta $$**: $$ \csc \theta = \frac{1}{\sin \theta} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4} $$ 5. **Find $$ \cot \theta $$**: $$ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{-2\sqrt{2}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{4} $$ ### Summary for Case 1: - $$ \sin \theta = \frac{2\sqrt{2}}{3} $$ - $$ \cos \theta = -\frac{1}{3} $$ - $$ \tan \theta = -2\sqrt{2} $$ - $$ \csc \theta = \frac{3\sqrt{2}}{4} $$ - $$ \sec \theta = 3 $$ - $$ \cot \theta = -\frac{\sqrt{2}}{4} $$ --- ### Case 2: $$ 15 \cdot \tan \theta = \frac{7}{3} $$, $$ \cos \theta < 0 $$, $$ \tan \theta > 0 $$ (Quadrant III) 1. **Calculate $$ \tan \theta $$**: $$ \tan \theta = \frac{7}{45} $$ 2. **Find $$ \sin \theta $$ and $$ \cos \theta $$**: Since $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$, we can express $$ \sin \theta $$ and $$ \cos \theta $$ in terms of a right triangle. Let $$ \sin \theta = 7k $$ and $$ \cos \theta = 45k $$ for some $$ k $$. 3. **Use the Pythagorean identity**: $$ \sin^2 \theta + \cos^2 \theta = 1 $$ $$ (7k)^2 + (45k)^2 = 1 $$ $$ 49k^2 + 2025k^2 = 1 $$ $$ 2074k^2 = 1 \implies k^2 = \frac{1}{2074} \implies k = \frac{1}{\sqrt{2074}} $$ 4. **Calculate $$ \sin \theta $$ and $$ \cos \theta $$**: $$ \sin \theta = 7k = \frac{7}{\sqrt{2074}}, \quad \cos \theta = 45k = \frac{45}{\sqrt{2074}} $$ Since both sine and cosine are negative in Quadrant III: $$ \sin \theta = -\frac{7}{\sqrt{2074}}, \quad \cos \theta = -\frac{45}{\sqrt{2074}} $$ 5. **Find $$ \csc \theta $$, $$ \sec \theta $$, and $$ \cot \theta $$**: $$ \csc \theta = \frac{1}{\sin \theta} = -\frac{\sqrt{2074}}{7} $$ $$ \sec \theta = \frac{1}{\cos \theta} = -\frac{\sqrt{2074}}{45} $$ $$ \cot \theta = \frac{1}{\tan \theta} = \frac{45}{7} $$ ### Summary for Case 2: - $$ \sin \theta = -\frac{7}{\sqrt{2074}} $$ - $$ \cos \theta = -\frac{45}{\sqrt{2074}} $$ - $$ \tan \theta = \frac{7}{45} $$ - $$ \csc \theta = -\frac{\sqrt{2074}}{7} $$ - $$ \sec \theta = -\frac{\sqrt{2074}}{45} $$ - $$ \cot \theta = \frac{45}{7} $$ These are the remaining five trigonometric functions for both cases. For the first case: - $$ \sin \theta = \frac{2\sqrt{2}}{3} $$ - $$ \cos \theta = -\frac{1}{3} $$ - $$ \tan \theta = -2\sqrt{2} $$ - $$ \csc \theta = \frac{3\sqrt{2}}{4} $$ - $$ \sec \theta = 3 $$ - $$ \cot \theta = -\frac{\sqrt{2}}{4} $$ For the second case: - $$ \sin \theta = -\frac{7}{\sqrt{2074}} $$ - $$ \cos \theta = -\frac{45}{\sqrt{2074}} $$ - $$ \tan \theta = \frac{7}{45} $$ - $$ \csc \theta = -\frac{\sqrt{2074}}{7} $$ - $$ \sec \theta = -\frac{\sqrt{2074}}{45} $$ - $$ \cot \theta = \frac{45}{7} $$

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

Upstudy AI Solution

Answer

Solution

Case 1: , , (Quadrant II)

Summary for Case 1:

Case 2: , , (Quadrant III)

Summary for Case 2:

Beyond the Answer

Related Questions

Latest Trigonometry Questions

Find the remaining five trigonometric functions of satisfying the conditions. (Hint: Draw a triangle in the appropriate quadrant. Pay attention to signs! 914. QII III

Upstudy AI Solution

Answer

Solution

Case 1: , , (Quadrant II)

Summary for Case 1:

Case 2: , , (Quadrant III)

Summary for Case 2:

Beyond the Answer

Related Questions

Latest Trigonometry Questions

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