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 = 3, c = 5 complete the table by using the names of the sides to express each trigonometric function as a ratio. sin b = csc b = cos b = sec b = tan b = cot b = the unknown side length b is (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) sin b = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) cos b = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) tan b = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find side $b$ via Pythagoras

For right triangle $ABC$ (right angle at $C$), Pythagorean theorem: $a^2 + b^2 = c^2$. Substitute $a=3, c=5$:
$$3^2 + b^2 = 5^2$$
$$9 + b^2 = 25$$
$$b^2 = 25 - 9 = 16$$
$$b = \sqrt{16} = 4$$

Step2: Define sides for $\angle B$

• Opposite to $\angle B$: $b=4$
• Adjacent to $\angle B$: $a=3$
• Hypotenuse: $c=5$

Step3: Calculate $\sin B$

Sine is opposite/hypotenuse:
$$\sin B = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}$$

Step4: Calculate $\cos B$

Cosine is adjacent/hypotenuse:
$$\cos B = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5}$$

Step5: Calculate $\tan B$

Tangent is opposite/adjacent:
$$\tan B = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}$$

Step6: Calculate $\csc B$

Cosecant is reciprocal of sine:
$$\csc B = \frac{1}{\sin B} = \frac{5}{4}$$

Step7: Calculate $\sec B$

Secant is reciprocal of cosine:
$$\sec B = \frac{1}{\cos B} = \frac{5}{3}$$

Step8: Calculate $\cot B$

Cotangent is reciprocal of tangent:
$$\cot B = \frac{1}{\tan B} = \frac{3}{4}$$

Answer:

The unknown side length $b$ is $\boldsymbol{4}$
$\sin B = \boldsymbol{\frac{4}{5}}$
$\cos B = \boldsymbol{\frac{3}{5}}$
$\tan B = \boldsymbol{\frac{4}{3}}$
$\csc B = \boldsymbol{\frac{5}{4}}$
$\sec B = \boldsymbol{\frac{5}{3}}$
$\cot B = \boldsymbol{\frac{3}{4}}$