Final answer:
The exact values of the six trigonometric functions depend on the specific angle in question. For a given angle in a right triangle, these functions can be determined using the lengths of the sides of the triangle or through the properties of specific triangle types.
Step-by-step explanation:
The exact values of the six trigonometric functions of an angle do indeed depend on the specific angles provided, meaning the correct answer is A) The exact values depend on the specific angles provided. In trigonometry, for a right triangle, the sine, cosine, and tangent functions are defined based on the opposite side, adjacent side, and the hypotenuse. The other three functions, cosecant, secant, and cotangent, are the reciprocals of sine, cosine, and tangent, respectively.
For example, in Figure 5.17, if x is the adjacent side, y is the opposite side, and h is the hypotenuse, the trigonometric functions are defined as follows:
- Sine (θ) = y/h
- Cosine (θ) = x/h
- Tangent (θ) = y/x
- Cosecant (θ) = h/y
- Secant (θ) = h/x
- Cotangent (θ) = x/y
These functions can have exact values that are obtained through either direct measurement, algebraic manipulation, or by using known identities and the properties of special triangles, such as 30°-60°-90° and 45°-45°-90° triangles. It is important when dealing with trigonometric calculations to ensure that the units of measurement for angles are consistent, typically in radians.