Question

2 select all of the true equations based on the figure.

a. \\( \cos(42) = \frac{b}{c} \\)

b. \\( \cos(48) = \frac{b}{c} \\)

c. \\( \sin(42) = \frac{b}{c} \\)

d. \\( \sin(48) = \frac{b}{c} \\)

e. \\( \tan(42) = \frac{b}{a} \\)

f. \\( \tan(48) = \frac{a}{b} \\)

3 while building a tent, jada ties a rope from the top of a pole 3 meters high to a stake that is 4 meters away from the base of the pole. jada draws this diagram to help find the angle made between the rope and the ground. which equation can jada use to find the value of \\( x \\)?

a. \\( x = \tan\left(\frac{3}{4}\
ight) \\)

b. \\( \tan(x) = \frac{3}{4} \\)

c. \\( x = \tan\left(\frac{4}{3}\
ight) \\)

d. \\( \tan(x) = \frac{4}{3} \\)

Explanation:

Question 2

1. In right triangle \( \triangle ABC \), \( \angle C = 90^\circ \), \( \angle A = 48^\circ \), so \( \angle B = 90^\circ - 48^\circ = 42^\circ \).
2. Recall trigonometric ratios: \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \), \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \).

• For \( \angle A = 48^\circ \): adjacent side is \( b \), hypotenuse is \( c \), so \( \cos(48^\circ)=\frac{b}{c} \) (B is true). Opposite side to \( 48^\circ \) is \( a \), so \( \sin(48^\circ)=\frac{a}{c} \), \( \tan(48^\circ)=\frac{a}{b} \) (F is true).
• For \( \angle B = 42^\circ \): adjacent side is \( a \), hypotenuse is \( c \), so \( \cos(42^\circ)=\frac{a}{c} \) (A is false). Opposite side to \( 42^\circ \) is \( b \), so \( \sin(42^\circ)=\frac{b}{c} \) (C is true). Adjacent side to \( 42^\circ \) is \( a \), opposite is \( b \), so \( \tan(42^\circ)=\frac{b}{a} \) (E is true).

In the right triangle (pole, ground, rope), the pole height (3 m) is the opposite side to angle \( x \), and the distance from stake to pole (4 m) is the adjacent side. Using the tangent ratio \( \tan(x)=\frac{\text{opposite}}{\text{adjacent}}=\frac{3}{4} \).

Answer:

B. \( \cos(48)=\frac{b}{c} \), C. \( \sin(42)=\frac{b}{c} \), E. \( \tan(42)=\frac{b}{a} \), F. \( \tan(48)=\frac{a}{b} \)

Question 3