Question 1: Using Cofunction Identities

We are given: $\sec(\phi) = \csc(42^\circ)$ Using the cofunction identity: $\sec(\phi) = \csc(90^\circ - \phi)$ This implies: $\phi = 90^\circ - 42^\circ$ Thus: $\phi = 48$ Answer for Question 1: 48

Question 2: Trigonometric Functions for Triangle

We have a right triangle with:

• Hypotenuse = 10
• One leg = 6 (adjacent to angle $\theta$)
• To find the opposite side, apply the Pythagorean theorem: $\text{Opposite side} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$

Now, compute the trigonometric functions:

1. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{10} = \frac{4}{5}$ ✔️ (Correct)

2. $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{10} = \frac{3}{5}$ ✔️ (Correct)

3. $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{6} = \frac{4}{3}$

   • The option says $\tan(\theta) = \frac{3}{4}$, which is incorrect ❌

4. $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{4}$

   • The option says $\csc(\theta) = \frac{4}{3}$, which is incorrect ❌

5. $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{10}{6} = \frac{5}{3}$

   • The option says $\sec(\theta) = \frac{5}{4}$, which is incorrect ❌

6. $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{3}{4}$ ✔️ (Correct)

✅ Correct Options:

• $\sin(\theta) = \frac{4}{5}$
• $\cos(\theta) = \frac{3}{5}$
• $\cot(\theta) = \frac{3}{4}$

Would you like more details or further explanations? 😊

5 Related Questions:

1. How are cofunction identities derived from right triangles?
2. Can you explain how to find missing triangle sides using trigonometry?
3. How do reciprocal trigonometric functions relate to primary functions?
4. What are the common mistakes when applying the Pythagorean theorem?
5. How do unit circle values help with trigonometric identities?

Tip:

When unsure about trigonometric ratios, quickly sketch the triangle to visualize opposite, adjacent, and hypotenuse relationships—it simplifies the process!