Question

suppose abc is a right triangle with sides a, b, and c and right angle at c. find the unknown side length using the pythagorean theorem and then find the values of the six trigonometric functions for angle b. a = 7, c = 25

Explanation:

Step1: Find side length b

By Pythagorean theorem $c^{2}=a^{2}+b^{2}$, so $b = \sqrt{c^{2}-a^{2}}$. Substituting $a = 7$ and $c = 25$, we have $b=\sqrt{25^{2}-7^{2}}=\sqrt{(25 + 7)(25 - 7)}=\sqrt{32\times18}=\sqrt{576}=24$.

Step2: Find sine of angle B

$\sin B=\frac{b}{c}$, substituting values gives $\sin B=\frac{24}{25}$.

Step3: Find cosine of angle B

$\cos B=\frac{a}{c}$, substituting values gives $\cos B=\frac{7}{25}$.

Step4: Find tangent of angle B

$\tan B=\frac{b}{a}$, substituting values gives $\tan B=\frac{24}{7}$.

Step5: Find cosecant of angle B

$\csc B=\frac{c}{b}$, substituting values gives $\csc B=\frac{25}{24}$.

Step6: Find secant of angle B

$\sec B=\frac{c}{a}$, substituting values gives $\sec B=\frac{25}{7}$.

Step7: Find cotangent of angle B

$\cot B=\frac{a}{b}$, substituting values gives $\cot B=\frac{7}{24}$.

Answer:

The unknown side length $b = 24$. The six - trigonometric functions for angle B are: $\sin B=\frac{24}{25}$, $\cos B=\frac{7}{25}$, $\tan B=\frac{24}{7}$, $\csc B=\frac{25}{24}$, $\sec B=\frac{25}{7}$, $\cot B=\frac{7}{24}$