Question

3.
3
x
20°
4.
5.
66°
x
12
7.
59°
x
16

Explanation:

Step1: Recall trigonometric ratios

For a right - triangle, the cosine of an angle $\theta$ is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, the tangent of an angle $\theta$ is defined as $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$, and the sine of an angle $\theta$ is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.

Problem 3:

Step1: Identify the trigonometric ratio

In the right - triangle, we know the adjacent side to the $20^{\circ}$ angle ($\text{adjacent}=3$) and we want to find the hypotenuse ($x$). Using the cosine ratio: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here $\theta = 20^{\circ}$, so $\cos(20^{\circ})=\frac{3}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{3}{\cos(20^{\circ})}$. Since $\cos(20^{\circ})\approx0.9397$, then $x=\frac{3}{0.9397}\approx3.19$.

Problem 5:

Step1: Identify the trigonometric ratio

In the right - triangle, we know the adjacent side to the $66^{\circ}$ angle ($\text{adjacent}=12$) and we want to find the opposite side ($x$). Using the tangent ratio: $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here $\theta = 66^{\circ}$, so $\tan(66^{\circ})=\frac{x}{12}$.

Step2: Solve for $x$

We know that $\tan(66^{\circ})\approx2.2460$, then $x = 12\times\tan(66^{\circ})$. So $x=12\times2.2460 = 26.95$.

Problem 7:

Step1: Identify the trigonometric ratio

In the right - triangle, we know the adjacent side to the $59^{\circ}$ angle ($\text{adjacent}=16$) and we want to find the hypotenuse ($x$). Using the cosine ratio: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here $\theta = 59^{\circ}$, so $\cos(59^{\circ})=\frac{16}{x}$.

Step2: Solve for $x$

We know that $\cos(59^{\circ})\approx0.5150$, then $x=\frac{16}{\cos(59^{\circ})}$. So $x=\frac{16}{0.5150}\approx31.07$.

Answer:

1. $x\approx3.19$; 5. $x\approx26.95$; 7. $x\approx31.07$