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.
\(\displaystyle{\sin{{t}}}=\frac{{3}}{{4}}{\quad\text{and}\quad}{\cos{{t}}}=\frac{\sqrt{{7}}}{{4}}\)

Answer 1

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, \(\displaystyle{\tan{{t}}}=\frac{{{\sin{{t}}}}}{{{\cos{{t}}}}}=\frac{{\frac{{3}}{{4}}}}{{\frac{\sqrt{{7}}}{{4}}}}=\frac{{3}}{\sqrt{{7}}}\)
\(\displaystyle{\csc{{t}}}=\frac{{1}}{{{\sin{{t}}}}}=\frac{{1}}{{\frac{{3}}{{4}}}}=\frac{{4}}{{3}}\)
\(\displaystyle{\sec{{t}}}=\frac{{1}}{{{\cos{{t}}}}}=\frac{{1}}{{\sqrt{\frac{3}{4}}}}=\frac{{4}}{\sqrt{{7}}}\)
\(\displaystyle{\cot{{t}}}=\frac{{{\cos{{t}}}}}{{{\sin{{t}}}}}=\frac{{\frac{\sqrt{{7}}}{{4}}}}{{\frac{{3}}{{4}}}}=\frac{\sqrt{{7}}}{{3}}\)