Question

evaluating trigonometric functions in exercises 5, 6, 7, 8, 9, and 10, find the exact values of the six trigonometric functions of the angle θ. 5. triangle with leg 8, leg 6, right angle 6. triangle with hypotenuse 13, leg 5, right angle show answer

Explanation:

Problem 5:

Step1: Find the hypotenuse

Using the Pythagorean theorem \( c = \sqrt{a^2 + b^2} \), where \( a = 6 \) and \( b = 8 \).
\( c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \)

Step2: Define opposite, adjacent, hypotenuse

For angle \( \theta \):

• Opposite side (\( \text{opp} \)) = 6
• Adjacent side (\( \text{adj} \)) = 8
• Hypotenuse (\( \text{hyp} \)) = 10

Step3: Calculate sine

\( \sin\theta = \frac{\text{opp}}{\text{hyp}} = \frac{6}{10} = \frac{3}{5} \)

Step4: Calculate cosine

\( \cos\theta = \frac{\text{adj}}{\text{hyp}} = \frac{8}{10} = \frac{4}{5} \)

Step5: Calculate tangent

\( \tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{6}{8} = \frac{3}{4} \)

Step6: Calculate cosecant

\( \csc\theta = \frac{1}{\sin\theta} = \frac{5}{3} \)

Step7: Calculate secant

\( \sec\theta = \frac{1}{\cos\theta} = \frac{5}{4} \)

Step8: Calculate cotangent

\( \cot\theta = \frac{1}{\tan\theta} = \frac{4}{3} \)

Step1: Find the other leg

Using the Pythagorean theorem \( b = \sqrt{c^2 - a^2} \), where \( c = 13 \) and \( a = 5 \).
\( b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \)

Step2: Define opposite, adjacent, hypotenuse

For angle \( \theta \):

• Opposite side (\( \text{opp} \)) = 5
• Adjacent side (\( \text{adj} \)) = 12
• Hypotenuse (\( \text{hyp} \)) = 13

Step3: Calculate sine

\( \sin\theta = \frac{\text{opp}}{\text{hyp}} = \frac{5}{13} \)

Step4: Calculate cosine

\( \cos\theta = \frac{\text{adj}}{\text{hyp}} = \frac{12}{13} \)

Step5: Calculate tangent

\( \tan\theta = \frac{\text{opp}}{\text{adj}} = \frac{5}{12} \)

Step6: Calculate cosecant

\( \csc\theta = \frac{1}{\sin\theta} = \frac{13}{5} \)

Step7: Calculate secant

\( \sec\theta = \frac{1}{\cos\theta} = \frac{13}{12} \)

Step8: Calculate cotangent

\( \cot\theta = \frac{1}{\tan\theta} = \frac{12}{5} \)

Answer:

\( \sin\theta = \frac{3}{5} \), \( \cos\theta = \frac{4}{5} \), \( \tan\theta = \frac{3}{4} \), \( \csc\theta = \frac{5}{3} \), \( \sec\theta = \frac{5}{4} \), \( \cot\theta = \frac{4}{3} \)

Problem 6: