Questions: Determine the exact values of the six trigonometric functions of the real number t. sin t= cos t= tan t= csc t= sec t= cot t=

Transcript text: Determine the exact values of the six trigonometric functions of the real number $t$. \[ \begin{array}{l} \sin t=\square \\ \cos t=\square \\ \tan t=\square \end{array} \] \[ \csc t= \] $\square$ \[ \sec t= \] $\square$ \[ \cot t= \] $\square$ Need Help? Watch It

Solution

Solution Steps

Step 1: Identify the coordinates

The coordinates given are $(-\frac{3}{5}, \frac{4}{5})$.

Step 2: Determine the values of sine and cosine

The sine of $ t $ is the y-coordinate: $\sin t = \frac{4}{5}$.
The cosine of $ t $ is the x-coordinate: $\cos t = -\frac{3}{5}$.

Step 3: Calculate the tangent

The tangent of $ t $ is the ratio of the sine to the cosine: \[ \tan t = \frac{\sin t}{\cos t} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3} \]

Step 4: Calculate the cosecant

The cosecant is the reciprocal of the sine: \[ \csc t = \frac{1}{\sin t} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]

Step 5: Calculate the secant

The secant is the reciprocal of the cosine: \[ \sec t = \frac{1}{\cos t} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \]

Step 6: Calculate the cotangent

The cotangent is the reciprocal of the tangent: \[ \cot t = \frac{1}{\tan t} = \frac{1}{-\frac{4}{3}} = -\frac{3}{4} \]

Final Answer

\[ \begin{align_} \sin t &= \frac{4}{5} \\ \cos t &= -\frac{3}{5} \\ \tan t &= -\frac{4}{3} \\ \csc t &= \frac{5}{4} \\ \sec t &= -\frac{5}{3} \\ \cot t &= -\frac{3}{4} \end{align_} \]

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