QuestionFind the six trigonometric functions for the point $\left(-\frac{1}{\sqrt{26}},-\frac{5}{\sqrt{26}}\right)$ on the unit circle.

Studdy Solution

STEP 1

Assumptions1. The point $\left(-\frac{1}{\sqrt{26}},-\frac{5}{\sqrt{26}}\right)$ lies on the unit circle. . The point corresponds to a real number $t$ .
3. We need to find the exact values of the six trigonometric functions of $t$ : $\sin(t)$ , $\cos(t)$ , $\tan(t)$ , $\csc(t)$ , $\sec(t)$ , and $\cot(t)$ .

STEP 2

On the unit circle, the x-coordinate corresponds to $\cos(t)$ and the y-coordinate corresponds to $\sin(t)$ . So we can directly read off the values of $\cos(t)$ and $\sin(t)$ from the given point.
$\cos(t) = -\frac{1}{\sqrt{26}}$ $\sin(t) = -\frac{5}{\sqrt{26}}$

STEP 3

The tangent of $t$ , $\tan(t)$ , is given by the ratio of the sine to the cosine $\tan(t) = \frac{\sin(t)}{\cos(t)}$

STEP 4

Substitute the values of $\sin(t)$ and $\cos(t)$ into the formula for $\tan(t)$ $\tan(t) = \frac{-\frac{}{\sqrt{26}}}{-\frac{1}{\sqrt{26}}}$

STEP 5

implify the expression for $\tan(t)$ $\tan(t) = \frac{5}{1} =5$

STEP 6

The cosecant of $t$ , $\csc(t)$ , is the reciprocal of the sine $\csc(t) = \frac{1}{\sin(t)}$

STEP 7

Substitute the value of $\sin(t)$ into the formula for $\csc(t)$ $\csc(t) = \frac{1}{-\frac{5}{\sqrt{26}}}$

STEP 8

implify the expression for $\csc(t)$ $\csc(t) = -\sqrt{26}$

STEP 9

The secant of $t$ , $\sec(t)$ , is the reciprocal of the cosine $\sec(t) = \frac{}{\cos(t)}$

STEP 10

Substitute the value of $\cos(t)$ into the formula for $\sec(t)$ $\sec(t) = \frac{}{-\frac{}{\sqrt{26}}}$

STEP 11

implify the expression for $\sec(t)$ $\sec(t) = -\sqrt{26}$

STEP 12

The cotangent of $t$ , $\cot(t)$ , is the reciprocal of the tangent $\cot(t) = \frac{}{\tan(t)}$

STEP 13

Substitute the value of $\tan(t)$ into the formula for $\cot(t)$ $\cot(t) = \frac{}{5}$ So, the six trigonometric functions of $t$ are $\sin(t) = -\frac{5}{\sqrt{26}}$ $\cos(t) = -\frac{}{\sqrt{26}}$ $\tan(t) =5$ $\csc(t) = -\sqrt{26}$ $\sec(t) = -\sqrt{26}$ $\cot(t) = \frac{}{5}$

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