For the angle shown is a point given on its terminal arm. [picture shows -12, -5 in quadrant three of the it circle].
Calculate the values of the three primary trigonmetric ratios, accurate to four decimal places.
To calculate the values of the three primary trigonometric ratios (sine, cosine, and tangent) for the given angle, you need to use the coordinates of the point on the terminal arm of the angle. In this case, the coordinates are (-12, -5) in the third quadrant of the unit circle.
To find the values of the trigonometric ratios, follow these steps:
1. Find the hypotenuse of the right triangle formed by the given point and the origin (0, 0). The hypotenuse is the distance from the origin to the point. Use the Pythagorean theorem:
hypotenuse = √((-12)^2 + (-5)^2)
= √(144 + 25)
= √(169)
= 13
2. Determine the values of the trigonometric ratios using the coordinates of the point and the hypotenuse:
- Sine (sin): sin(theta) = opposite/hypotenuse = -5/13
- Cosine (cos): cos(theta) = adjacent/hypotenuse = -12/13
- Tangent (tan): tan(theta) = opposite/adjacent = -5/-12 = 5/12
To round these values to four decimal places:
- Sine (sin): -5/13 ≈ -0.3846
- Cosine (cos): -12/13 ≈ -0.9231
- Tangent (tan): 5/12 ≈ 0.4167
Therefore, the values of the three primary trigonometric ratios for the given angle are approximately:
- sin(theta) ≈ -0.3846
- cos(theta) ≈ -0.9231
- tan(theta) ≈ 0.4167