Question

If tan(θ) = 3/4, find all the other ratios of θ.

A) sin(θ) = 3/5, cos(θ) = 4/5, csc(θ) = 5/3, sec(θ) = 5/4, cot(θ) = 4/3
B) sin(θ) = 3/5, cos(θ) = 4/5, csc(θ) = 5/3, sec(θ) = 4/5, cot(θ) = 4/3
C) sin(θ) = 4/5, cos(θ) = 3/5, csc(θ) = 5/4, sec(θ) = 5/3, cot(θ) = 3/4
D) sin(θ) = 4/5, cos(θ) = 3/5, csc(θ) = 5/4, sec(θ) = 3/5, cot(θ) = 3/4

Answer 1

Given tan(θ) = 3/4, the other trigonometric ratios are calculated based on a right-angled triangle with sides 3, 4, and 5. The correct ratios are sin(θ) = 3/5, cos(θ) = 4/5, csc(θ) = 5/3, sec(θ) = 5/4, and cot(θ) = 4/3.

Step-by-step explanation:

If tan(θ) = 3/4, we are dealing with a right triangle where the opposite side (the length related to the sin function) is 3 units and the adjacent side (the length related to the cos function) is 4 units. To find the hypotenuse, we apply the Pythagorean theorem, so hypotenuse = √(32 + 42) = 5. Now we can find all the trigonometric ratios for θ:

• sin(θ) = opposite/hypotenuse = 3/5
• cos(θ) = adjacent/hypotenuse = 4/5
• csc(θ) = 1/sin(θ) = 5/3
• sec(θ) = 1/cos(θ) = 5/4
• cot(θ) = 1/tan(θ) = 4/3

Thus, the correct answer is A) sin(θ) = 3/5, cos(θ) = 4/5, csc(θ) = 5/3, sec(θ) = 5/4, cot(θ) = 4/3.