Questions: A right triangle with acute angle θ is given. Evaluate the six trigonometric functions of the acute angle θ.

Solution Steps

To evaluate the six trigonometric functions of an acute angle \(\theta\) in a right triangle, we need to know the lengths of the sides of the triangle. Let's assume the lengths of the opposite side, adjacent side, and hypotenuse are given as opposite, adjacent, and hypotenuse, respectively. The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent, which can be calculated using these side lengths.

In a right triangle, we are given the side lengths: the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5.

The sine of angle \(\theta\) is calculated as the ratio of the length of the opposite side to the hypotenuse: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6000 \]

The cosine of angle \(\theta\) is calculated as the ratio of the length of the adjacent side to the hypotenuse: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8000 \]

The tangent of angle \(\theta\) is calculated as the ratio of the length of the opposite side to the adjacent side: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} = 0.7500 \]

The cosecant of angle \(\theta\) is the reciprocal of the sine: \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3} \approx 1.6667 \]

The secant of angle \(\theta\) is the reciprocal of the cosine: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{4} = 1.2500 \]

The cotangent of angle \(\theta\) is the reciprocal of the tangent: \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{4}{3} \approx 1.3333 \]

Final Answer