Question

find the exact values of six trigonometric ratios for the mentioned angle.

1. ∠r

sin = \frac{o}{h}=
cos = \frac{a}{h}=
tan = \frac{o}{a}=

1. ∠l

sin =
cos =
tan =

1. ∠b

sin =
cos =
tan =

Explanation:

Step1: Recall trigonometric ratios

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 \(\angle R\) in \(\triangle RTS\):

• Opposite side to \(\angle R\): \(ST = 7\)
• Adjacent side to \(\angle R\): \(RT=24\)
• Hypotenuse: \(RS = 25\)
• \(\sin R=\frac{ST}{RS}=\frac{7}{25}\)
• \(\cos R=\frac{RT}{RS}=\frac{24}{25}\)
• \(\tan R=\frac{ST}{RT}=\frac{7}{24}\)

For \(\angle L\) in \(\triangle JKL\):

• Opposite side to \(\angle L\): \(JK = 15\)
• Adjacent side to \(\angle L\): \(KL = 8\)
• Hypotenuse: \(JL = 17\)
• \(\sin L=\frac{JK}{JL}=\frac{15}{17}\)
• \(\cos L=\frac{KL}{JL}=\frac{8}{17}\)
• \(\tan L=\frac{JK}{KL}=\frac{15}{8}\)

For \(\angle B\) in \(\triangle ABC\):

• Opposite side to \(\angle B\): \(AC = 15\)
• Adjacent side to \(\angle B\): \(AB = 8\)
• Hypotenuse: \(BC = 17\)
• \(\sin B=\frac{AC}{BC}=\frac{15}{17}\)
• \(\cos B=\frac{AB}{BC}=\frac{8}{17}\)
• \(\tan B=\frac{AC}{AB}=\frac{15}{8}\)

Answer:

1. \(\sin R=\frac{7}{25}\), \(\cos R=\frac{24}{25}\), \(\tan R=\frac{7}{24}\)
2. \(\sin L=\frac{15}{17}\), \(\cos L=\frac{8}{17}\), \(\tan L=\frac{15}{8}\)
3. \(\sin B=\frac{15}{17}\), \(\cos B=\frac{8}{17}\), \(\tan B=\frac{15}{8}\)