Questions: Find the exact values of the six trigonometric functions of the real number t. sin t=0.724 cos t=0.690 tan t=1.05 csc t=1.45 sec t=1.381 cot t=0.952

Solution Steps

The coordinates given are \(\left(\frac{21}{29}, \frac{20}{29}\right)\).

The radius \(r\) of the circle can be calculated using the Pythagorean theorem: \[ r = \sqrt{\left(\frac{21}{29}\right)^2 + \left(\frac{20}{29}\right)^2} \] \[ r = \sqrt{\frac{441}{841} + \frac{400}{841}} \] \[ r = \sqrt{\frac{841}{841}} \] \[ r = 1 \]

Using the coordinates and the radius, we can find the six trigonometric functions:

1. Sine: \[ \sin t = \frac{y}{r} = \frac{20}{29} \]

2. Cosine: \[ \cos t = \frac{x}{r} = \frac{21}{29} \]

3. Tangent: \[ \tan t = \frac{y}{x} = \frac{\frac{20}{29}}{\frac{21}{29}} = \frac{20}{21} \]

4. Cosecant: \[ \csc t = \frac{1}{\sin t} = \frac{1}{\frac{20}{29}} = \frac{29}{20} \]

5. Secant: \[ \sec t = \frac{1}{\cos t} = \frac{1}{\frac{21}{29}} = \frac{29}{21} \]

6. Cotangent: \[ \cot t = \frac{1}{\tan t} = \frac{1}{\frac{20}{21}} = \frac{21}{20} \]

Final Answer