Question

7.5.8 check your understanding
right triangle sbg is shown, where gs = 13 and sb = 12.
select all the true statements.
□ \\(\cos (\angle gsb) = \frac{12}{13}\\)
□ \\(\sin(\angle gsb) = \frac{5}{13}\\)
□ \\(\cos(\angle gsb) = \frac{5}{13}\\)
□ \\(\tan(\angle gsb) = \frac{5}{12}\\)
□ \\(\tan(\angle gsb) = \frac{12}{5}\\)

Explanation:

Step1: Find the length of GB

In right triangle \( SBG \), by the Pythagorean theorem \( GS^{2}=SB^{2}+GB^{2} \). Given \( GS = 13 \), \( SB=12 \), so \( GB=\sqrt{GS^{2}-SB^{2}}=\sqrt{13^{2}-12^{2}}=\sqrt{169 - 144}=\sqrt{25} = 5 \).

Step2: Analyze \( \cos(\angle GSB) \)

For angle \( \angle GSB \), adjacent side to \( \angle GSB \) is \( SB = 12 \), hypotenuse is \( GS=13 \). By definition of cosine (\( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \)), \( \cos(\angle GSB)=\frac{SB}{GS}=\frac{12}{13} \), so this statement is true.

Step3: Analyze \( \sin(\angle GSB) \)

For angle \( \angle GSB \), opposite side to \( \angle GSB \) is \( GB = 5 \), hypotenuse is \( GS = 13 \). By definition of sine (\( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \)), \( \sin(\angle GSB)=\frac{GB}{GS}=\frac{5}{13} \), so this statement is true.

Step4: Analyze \( \cos(\angle GSB)=\frac{5}{13} \)

From Step2, we know \( \cos(\angle GSB)=\frac{12}{13}
eq\frac{5}{13} \), so this statement is false.

Step5: Analyze \( \tan(\angle GSB) \)

For angle \( \angle GSB \), opposite side is \( GB = 5 \), adjacent side is \( SB = 12 \). By definition of tangent (\( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \)), \( \tan(\angle GSB)=\frac{GB}{SB}=\frac{5}{12} \), so \( \tan(\angle GSB)=\frac{5}{12} \) is true, and \( \tan(\angle GSB)=\frac{12}{5} \) is false.

Answer:

• \( \boldsymbol{\cos (\angle GSB) = \frac{12}{13}} \) (True)
• \( \boldsymbol{\sin(\angle GSB) = \frac{5}{13}} \) (True)
• \( \boldsymbol{\tan(\angle GSB) = \frac{5}{12}} \) (True)