Questions: The point given below is on the terminal side of an angle θ. Find the exact value of each of the six trigonometric functions of θ. (-8,6)

Solution Steps

To find the hypotenuse \( r \) of the right triangle formed by the point \((-8, 6)\), we use the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-8)^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

The sine of the angle \( \theta \) is given by: \[ \sin(\theta) = \frac{y}{r} = \frac{6}{10} = \frac{3}{5} \] Thus, the exact value is: \[ \sin(\theta) = \frac{3}{5} \]

The cosine of the angle \( \theta \) is given by: \[ \cos(\theta) = \frac{x}{r} = \frac{-8}{10} = -\frac{4}{5} \] Thus, the exact value is: \[ \cos(\theta) = -\frac{4}{5} \]

The tangent of the angle \( \theta \) is given by: \[ \tan(\theta) = \frac{y}{x} = \frac{6}{-8} = -\frac{3}{4} \] Thus, the exact value is: \[ \tan(\theta) = -\frac{3}{4} \]

The cosecant is the reciprocal of sine: \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \] Thus, the exact value is: \[ \csc(\theta) = \frac{5}{3} \]

The secant is the reciprocal of cosine: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} \] Thus, the exact value is: \[ \sec(\theta) = -\frac{5}{4} \]

The cotangent is the reciprocal of tangent: \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3} \] Thus, the exact value is: \[ \cot(\theta) = -\frac{4}{3} \]

Final Answer