Question

In a right-angle triangle, if Tan(θ) = 3/4, what is cos(θ) - sin(θ)?

A) 1/7
B) 3/5
C) 1/5
D) 2/5

Answer 1

Given that tan(θ) = 3/4, we find the hypotenuse of the right-angled triangle using the Pythagorean theorem and then calculate cos(θ) and sin(θ). Subtracting sin(θ) from cos(θ), the solution is 1/5.

Step-by-step explanation:

The student has asked about finding the value of cos(θ) - sin(θ) given that tan(θ) = 3/4 in a right-angle triangle. To find cos(θ) and sin(θ), we can utilize the Pythagorean identity, which states that sin(θ) and cos(θ) are related by the equation sin2(θ) + cos2(θ) = 1.

Since tan(θ) = opposite/adjacent and we have a right-angled triangle, we know:

• Opposite side length = 3 units
• Adjacent side length = 4 units

Using the Pythagorean theorem, we can find the hypotenuse:

hypotenuse2 = opposite2 + adjacent2 = 32 + 42 = 9 + 16 = 25

Hence, the hypotenuse = √25 = 5 units.

So, cos(θ) = adjacent/hypotenuse = 4/5 and sin(θ) = opposite/hypotenuse = 3/5.

Now, we calculate cos(θ) - sin(θ) = (4/5) - (3/5) = 1/5.

The correct answer is C) 1/5.