Question

find the sine, cosine, and tangent of ∠a.
a right triangle abc with right angle at b. the length of ab is 90, the length of bc is 72.
simplify your answers and write them as proper fractions, improper fractions, or whole numbers.
sin(a) =
cos(a) =
tan(a) =
submit

Explanation:

Step1: Find side AB

First, use the Pythagorean theorem for $\triangle ABC$:
$$AB = \sqrt{AC^2 - BC^2} = \sqrt{90^2 - 72^2}$$
$$=\sqrt{8100 - 5184}=\sqrt{2916}=54$$

Step2: Calculate $\sin(A)$

$\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AC}$
$$\sin(A)=\frac{72}{90}=\frac{4}{5}$$

Step3: Calculate $\cos(A)$

$\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AB}{AC}$
$$\cos(A)=\frac{54}{90}=\frac{3}{5}$$

Step4: Calculate $\tan(A)$

$\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AB}$
$$\tan(A)=\frac{72}{54}=\frac{4}{3}$$

Answer:

$\sin(A)=\frac{4}{5}$, $\cos(A)=\frac{3}{5}$, $\tan(A)=\frac{4}{3}$