Question

se the figure below to find the following:
a. sin a
b. cos a
c. tan a
d. csc a
e. sec a
f. cot a
(figure: right triangle with right angle at c, bc = 5 ft, ac = 12 ft, ab = 13 ft)

Explanation:

Step1: Identify sides for angle A

In right triangle \( \triangle ABC \), \( \angle C = 90^\circ \), hypotenuse \( AB = 13 \) ft, opposite side to \( A \) is \( BC = 5 \) ft, adjacent side to \( A \) is \( AC = 12 \) ft.

Step2: Calculate \( \sin A \)

\( \sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AB}=\frac{5}{13} \)

Step3: Calculate \( \cos A \)

\( \cos A=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AC}{AB}=\frac{12}{13} \)

Step4: Calculate \( \tan A \)

\( \tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}=\frac{5}{12} \)

Step5: Calculate \( \csc A \)

\( \csc A=\frac{1}{\sin A}=\frac{13}{5} \)

Step6: Calculate \( \sec A \)

\( \sec A=\frac{1}{\cos A}=\frac{13}{12} \)

Step7: Calculate \( \cot A \)

\( \cot A=\frac{1}{\tan A}=\frac{12}{5} \)

Answer:

a. \( \sin A = \boldsymbol{\frac{5}{13}} \)
b. \( \cos A = \boldsymbol{\frac{12}{13}} \)
c. \( \tan A = \boldsymbol{\frac{5}{12}} \)
d. \( \csc A = \boldsymbol{\frac{13}{5}} \)
e. \( \sec A = \boldsymbol{\frac{13}{12}} \)
f. \( \cot A = \boldsymbol{\frac{12}{5}} \)