Question

If (tan(θ) = 3) and (0 < θ < 90^∘), evaluate the following:

a) (sin(θ))
b) (cos(θ))

Answer 1

The student's question about evaluating sin(θ) and cos(θ) where tan(θ) = 3 is solved by creating a right triangle and using the definitions of sine and cosine in relation to the sides of the triangle.

Step-by-step explanation:

If we have tan(θ) = 3 and the angle θ is in the first quadrant (0 < θ < 90 degrees), we can find sin(θ) and cos(θ) by interpreting the tangent as the ratio of the opposite side to the adjacent side in a right triangle. We create a right triangle where the opposite side (a) is 3 units and the adjacent side (b) is 1 unit. Using the Pythagorean theorem, the hypotenuse (c) will then be √(32 + 12) = √10.

Therefore, sin(θ) is the ratio of the opposite side to the hypotenuse which is a/c or 3/√10, which simplifies to 3√10/10 after rationalizing the denominator.

The cos(θ) is the ratio of the adjacent side to the hypotenuse which is b/c or 1/√10, which simplifies to √10/10 after rationalizing the denominator.