Question

question. in right triangle △abc, the right - angle is at c, and tan b = 4/3. what is the value of cos a? a 4/3 b 3/4 c 5/4 d 4/5

Explanation:

Step1: Recall tangent - side relationship

In right - triangle $\triangle ABC$ with right - angle at $C$, $\tan B=\frac{AC}{BC}=\frac{4}{3}$. Let $AC = 4x$ and $BC = 3x$.

Step2: Use Pythagorean theorem

By the Pythagorean theorem $AB^{2}=AC^{2}+BC^{2}$. Substituting $AC = 4x$ and $BC = 3x$, we get $AB^{2}=(4x)^{2}+(3x)^{2}=16x^{2}+9x^{2}=25x^{2}$, so $AB = 5x$.

Step3: Find $\cos A$

The cosine of an angle in a right - triangle is defined as $\cos A=\frac{AC}{AB}$. Since $AC = 4x$ and $AB = 5x$, then $\cos A=\frac{4}{5}$.

Answer:

A. $\frac{4}{5}$