The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ.
Given point: (7, -6)
To find the exact values of the six trigonometric functions of θ, we need to determine the values of cosine, sine, tangent, cotangent, secant, and cosecant.
1. Start by drawing a right-angled triangle with the given point as one of its vertices (the end of the terminal side).
2. Use the given point (7, -6) to find the lengths of the sides of the triangle. The x-coordinate (7) represents the adjacent side, and the y-coordinate (-6) represents the opposite side.
3. To find the hypotenuse of the triangle, use the Pythagorean theorem: a^2 + b^2 = c^2. Substitute the values of the adjacent side (7) and the opposite side (-6) into the equation and solve for c. We get 7^2 + (-6)^2 = c^2, which simplifies to 49 + 36 = c^2. Therefore, c^2 = 85, and c ≈ 9.22 (approximated to two decimal places).
4. With the lengths of the sides determined, you can now find the trigonometric functions of θ:
- Cosine (cos θ) is equal to the ratio of the adjacent side to the hypotenuse. So, cos θ = 7/9.22 ≈ 0.759 (approximated to three decimal places).
- Sine (sin θ) is equal to the ratio of the opposite side to the hypotenuse. So, sin θ = -6/9.22 ≈ -0.651 (approximated to three decimal places).
- Tangent (tan θ) is equal to the ratio of the opposite side to the adjacent side. So, tan θ = -6/7 ≈ -0.857 (approximated to three decimal places).
- Cotangent (cot θ) is equal to the reciprocal of the tangent. So, cot θ = 1/tan θ = 1/(-0.857) ≈ -1.167 (approximated to three decimal places).
- Secant (sec θ) is equal to the reciprocal of the cosine. So, sec θ = 1/cos θ = 1/0.759 ≈ 1.318 (approximated to three decimal places).
- Cosecant (csc θ) is equal to the reciprocal of the sine. So, csc θ = 1/sin θ = 1/(-0.651) ≈ -1.536 (approximated to three decimal places).
Therefore, the exact values of the six trigonometric functions of θ are:
cos θ ≈ 0.759
sin θ ≈ -0.651
tan θ ≈ -0.857
cot θ ≈ -1.167
sec θ ≈ 1.318
csc θ ≈ -1.536