Question

Given tan(θ)= 16/63, determine sin(θ), sec(θ), and cot(θ).

A) sin(θ)= 4/5, sec(θ)= 5/3, cot(θ)= 3/4
B) sin(θ)= 63/65, sec(θ)= 65/16, cot(θ)= 16/63
C) sin(θ)= 63/65, sec(θ)= 65/63, cot(θ)= 63/16
D) sin(θ) =63/65,sec(θ)=65/16, ( \

Answer 1

To find sin(θ), sec(θ), and cot(θ) given tan(θ)= 16/63, we use the Pythagorean theorem to find the hypotenuse of the right-angled triangle and then apply the definitions of sine, secant and cotangent to find the corresponding ratios.

Step-by-step explanation:

Given that tan(θ)= 16/63, we need to find sin(θ), sec(θ), and cot(θ). We will use the definitions of trigonometric functions based on a right-angled triangle. In a right-angled triangle, the tangent of an angle θ, which is opposite/adjacent, matches the given tan(θ).

• Let the opposite side be 16 (matching the numerator of tan(θ)),
  
• and the adjacent side be 63 (matching the denominator of tan(θ)).

We need to find the hypotenuse using the Pythagorean theorem:

hypotenuse = √(opposite² + adjacent²) = √(16² + 63²) = √(256 + 3969) = √(4225) = 65.

Therefore, sin(θ) is opposite/hypotenuse = 16/65, sec(θ) is hypotenuse/adjacent = 65/63, and cot(θ) is adjacent/opposite = 63/16. The correct answer is D) sin(θ) = 16/65, sec(θ) = 65/63, cot(θ) = 63/16.

To find sec(θ), we can use the fact that sec(θ) = 1/cos(θ). We can find cos(θ) from tan(θ) as cos(θ) = 1/√(1+tan²(θ)). Finally, to find cot(θ), we can use the fact that cot(θ) = 1/tan(θ). Therefore, sin(θ) = 16/√(16²+63²), sec(θ) = 1/√(1+tan²(θ)), and cot(θ) = 1/tan(θ).