Question

For a standard position angle determined by the point (x,y) what are the values of the trigonometric functions?. For the point (16,12) find sin theta and cos theta

Answer 1

For the point (16,12), the sine (sinθ) is 0.6 and the cosine (cosθ) is 0.8. These are calculated using the Pythagorean theorem to find the hypotenuse and then dividing the opposite and adjacent sides by the hypotenuse, respectively.

Step-by-step explanation:

The value of the trigonometric functions for a point in standard position can be determined using the definitions of sine (sin) and cosine (cos) in a right triangle. For the point (16,12) in standard position, we can consider this point as lying on a right-angled triangle where 16 is the length of the adjacent side to the angle (x-coordinate), 12 is the length of the opposite side to the angle (y-coordinate), and the hypotenuse can be calculated using the Pythagorean theorem.

Calculating the hypotenuse (denoted as 'h'), we get:

• h = √(x² + y²) = √(16² + 12²)
• h = √(256 + 144)
• h = √400
• h = 20

Now using the definitions of sinθ and cosθ:

• sinθ = opposite/hypotenuse = 12/20 = 0.6
• cosθ = adjacent/hypotenuse = 16/20 = 0.8

Therefore, for the point (16,12), sinθ is 0.6 and cosθ is 0.8.

Answer 2

First we must find the hypotenuse:
c² = x² + y²
c² = 16² + 12²
c² = 256 + 144
c² = 400
c = √ 400
c = 20
sin t(theta) = y / c = 12/20 = 3 / 5
cos t(theta) = x / c = 16 / 20 = 4 / 5