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=

Solution Steps

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

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

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

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

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

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

Final Answer