Question

right triangle dfr is shown, where fd = 3 and rd = 4.
select all the true statements.
□ \\(\cos(\angle frd) = \frac{3}{5}\\)
□ \\(\sin(\angle frd) = \frac{3}{5}\\)
□ \\(\cos(\angle frd) = \frac{4}{5}\\)
□ \\(\tan(\angle frd) = \frac{3}{4}\\)
□ \\(\sin(\angle frd) = \frac{3}{4}\\)

Explanation:

Step1: Find the hypotenuse \( FR \)

Using the Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( a = 3 \), \( b = 4 \). So \( FR=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5 \).

Step2: Recall trigonometric ratios

For an acute angle \( \theta \) in a right triangle, \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \), \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \).
For \( \angle FRD \):

• Opposite side to \( \angle FRD \) is \( FD = 3 \)
• Adjacent side to \( \angle FRD \) is \( RD = 4 \)
• Hypotenuse \( FR = 5 \)

Step3: Evaluate each option

• \( \cos(\angle FRD)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{4}{5}\), so \( \cos(\angle FRD)=\frac{3}{5} \) is false, \( \cos(\angle FRD)=\frac{4}{5} \) is true.
• \( \sin(\angle FRD)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{3}{5}\), so \( \sin(\angle FRD)=\frac{3}{5} \) is true, \( \sin(\angle FRD)=\frac{3}{4} \) is false.
• \( \tan(\angle FRD)=\frac{\text{opposite}}{\text{adjacent}}=\frac{3}{4}\), so \( \tan(\angle FRD)=\frac{3}{4} \) is true.

Answer:

B. \( \sin(\angle FRD)=\frac{3}{5} \)
C. \( \cos(\angle FRD)=\frac{4}{5} \)
D. \( \tan(\angle FRD)=\frac{3}{4} \)