Question

a point p(x,y) is shown on the unit circle corresponding to a real number t. find the values of the trigonometric functions at t.
b. cos t = $\frac{sqrt{2}}{2}$ (type an exact answer, using radicals as needed. simplify your answer and rationalize the denominator.)
c. tan t = 1 (type an exact answer, using radicals as needed. simplify your answer and rationalize the denominator.)
d. csc t = $sqrt{2}$ (type an exact answer, using radicals as needed. simplify your answer and rationalize the denominator.)
e. sec t = (type an exact answer, using radicals as needed. simplify your answer and rationalize the denominator.)

Explanation:

Step1: Recall secant - cosine relationship

$\sec t=\frac{1}{\cos t}$

Step2: Substitute the value of $\cos t$

Given $\cos t = \frac{\sqrt{2}}{2}$, then $\sec t=\frac{1}{\frac{\sqrt{2}}{2}}$.

Step3: Simplify the expression

$\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}$. Rationalize the denominator: $\frac{2}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\sqrt{2}$.

Answer:

$\sqrt{2}$