Question

# Given the values for sin t and cos t, use reciprocal and quotient identities to find the values of the other trigonometric functions of t. sin t = 3/4 and cos t=sqrt7/4

Trigonometric Functions
Given the values for sin t and cos t, use reciprocal and quotient identities to find the values of the other trigonometric functions of t.
$$\displaystyle{\sin{{t}}}=\frac{{3}}{{4}}{\quad\text{and}\quad}{\cos{{t}}}=\frac{\sqrt{{7}}}{{4}}$$

2021-03-07

Consider the given trigonometric function sint and cost.
The other trigonometric functions values are find as,
Use the trigonometric formula,
$$\displaystyle{\tan{{t}}}=\frac{{{\sin{{t}}}}}{{{\cos{{t}}}}}$$
$$\displaystyle{\csc{{t}}}=\frac{{1}}{{{\sin{{t}}}}}$$
$$\displaystyle{\sec{{t}}}=\frac{{1}}{{{\cos{{t}}}}}$$
$$\displaystyle{\cot{{t}}}=\frac{{{\cos{{t}}}}}{{{\sin{{t}}}}}$$
Now, substitute the given trigonometric function values in the above formula,
Since $$\displaystyle{\sin{{t}}}=\frac{{3}}{{4}}{\quad\text{and}\quad}{\cos{{t}}}=\frac{\sqrt{{7}}}{{4}}$$
Thus, $$\displaystyle{\tan{{t}}}=\frac{{{\sin{{t}}}}}{{{\cos{{t}}}}}=\frac{{\frac{{3}}{{4}}}}{{\frac{\sqrt{{7}}}{{4}}}}=\frac{{3}}{\sqrt{{7}}}$$
$$\displaystyle{\csc{{t}}}=\frac{{1}}{{{\sin{{t}}}}}=\frac{{1}}{{\frac{{3}}{{4}}}}=\frac{{4}}{{3}}$$
$$\displaystyle{\sec{{t}}}=\frac{{1}}{{{\cos{{t}}}}}=\frac{{1}}{{\sqrt{\frac{3}{4}}}}=\frac{{4}}{\sqrt{{7}}}$$
$$\displaystyle{\cot{{t}}}=\frac{{{\cos{{t}}}}}{{{\sin{{t}}}}}=\frac{{\frac{\sqrt{{7}}}{{4}}}}{{\frac{{3}}{{4}}}}=\frac{\sqrt{{7}}}{{3}}$$