To find the values of the six trigonometric functions of angle θ, which is in standard position and has the point (5, −2) on its terminal ray, you would need to use the following steps:
- First, you would need to find the length of the hypotenuse of the right triangle formed by the point (5, −2), the origin, and the projection of the point on the x-axis. You can use the Pythagorean theorem to do this: r^2 = x^2 + y^2, where r is the hypotenuse, x is the horizontal side, and y is the vertical side. In this case, x = 5 and y = −2, so r^2 = 5^2 + (−2)^2 = 25 + 4 = 29. Taking the square root of both sides, you get r = √29.
- Second, you would need to use the definitions of the trigonometric functions in terms of the sides of the right triangle. The six trigonometric functions are: sine, cosine, tangent, cosecant, secant, and cotangent. They are defined as follows:
- sin(θ) = y/r
- cos(θ) = x/r
- tan(θ) = y/x
- csc(θ) = r/y
- sec(θ) = r/x
- cot(θ) = x/y
- Third, you would need to substitute the values of x, y, and r into the definitions and simplify if possible. For example, sin(θ) = y/r = −2/√29. You can rationalize the denominator by multiplying both numerator and denominator by √29. This gives you sin(θ) = −2√29/29. You can repeat this process for the other five functions.
This is how you would find the values of the six trigonometric functions of angle θ without evaluating them. I hope this helps you understand how to use the right triangle approach to find trigonometric function values. If you want to learn more about this topic, you can check out some of these resources:
- [Trigonometric Functions of Any Angle | Precalculus II](^1^)
- [Trigonometry: Right Triangle Approach | Math Is Fun](^2^)
- [Trigonometric Functions - Math Open Reference](^3^)