Let's break down the key elements in your text and go through each mathematical problem presented:

Problem 1:

You are asked to find all values of $\theta$ in the interval $[0^\circ, 360^\circ)$ such that $\cos(\theta) = \frac{1}{2}$. The answer provided is D: $60^\circ$ and $300^\circ$.

Explanation:

• The cosine function is positive in Quadrants I and IV.
• $\cos(60^\circ) = \frac{1}{2}$, and in Quadrant IV, the angle with the same reference angle of $60^\circ$ is $360^\circ - 60^\circ = 300^\circ$. Thus, the angles are $60^\circ$ and $300^\circ$.

Problem 2:

You're asked to find the value of $\theta$ in the interval $[0^\circ, 90^\circ]$ such that $\sec(\theta) = 2.165$, and the answer is C: $62.4906237^\circ$.

Explanation:

• $\sec(\theta) = 2.165$ means that $\cos(\theta) = \frac{1}{2.165}$.
• Using a calculator, $\theta = \cos^{-1}\left(\frac{1}{2.165}\right) \approx 62.4906237^\circ$.

Problem 3:

This problem involves solving a right triangle where $a = 3.4 \, \text{cm}$, $b = 2.5 \, \text{cm}$, and $\angle C = 90^\circ$. The correct solution is A: $\angle A = 53.673^\circ$, $\angle B = 36.327^\circ$, and $c = 4.220 \, \text{cm}$.

Explanation:

• Use the Pythagorean theorem to find the hypotenuse: $c = \sqrt{a^2 + b^2} = \sqrt{3.4^2 + 2.5^2} = \sqrt{11.56 + 6.25} = \sqrt{17.81} \approx 4.220 \, \text{cm}.$
• Find angles using trigonometry: $\tan(A) = \frac{a}{b} = \frac{3.4}{2.5}, \quad A = \tan^{-1}\left(\frac{3.4}{2.5}\right) \approx 53.673^\circ.$ Then, $B = 90^\circ - A = 90^\circ - 53.673^\circ = 36.327^\circ$.

Extra Steps for Reference Angle in Problem 1:

• Cosine is positive in Quadrant I and IV.
• In Quadrant I, $\theta = 60^\circ$.
• In Quadrant IV, subtract the reference angle from $360^\circ$: $360^\circ - 60^\circ = 300^\circ$.

Would you like more details on any of these steps?

Here are 5 related questions to further your understanding:

1. What are the general solutions for $\cos(\theta) = \frac{1}{2}$ in radians?
2. How do you find the reference angle for a given trigonometric value?
3. How can you derive the secant function's inverse on a calculator?
4. What is the general rule for solving a right triangle given two sides?
5. How can you find angles using sine or cosine in a non-right triangle?

Tip:

For trigonometric problems, always remember which quadrants the sine, cosine, and tangent functions are positive in: All Students Take Calculus (ASTC) for Quadrants I, II, III, and IV respectively.