Question

Given cos 2θ = 3/5 and cos θ terminates in quadrant 1, find sin θ and cos θ.

a) sin θ = 4/5, cos θ = 3/5
b) sin θ = 3/5, cos θ = 4/5
c) sin θ = -4/5, cos θ = -3/5
d) sin θ = -3/5, cos θ = -4/5

Answer 1

The correct values for sin θ and cos θ, given that cos 2θ = 3/5 are sin θ = 4/5 and cos θ = 3/5, which corresponds to quadrant 1 where both sine and cosine are positive. The correct option is a.

Step-by-step explanation:

To find the values of sin θ and cos θ given that cos 2θ = 3/5 and cos θ is in quadrant 1, we use trigonometric identities and the properties of a right-angled triangle.

Using the double angle formula for cosine, cos 2θ = cos² θ - sin² θ, we can plug in cos 2θ = 3/5 and find two possible values for cos² θ. However, since cos θ is positive in quadrant 1, only the positive root is valid for cos θ. We can then express sin θ in terms of cos θ using the Pythagorean identity sin² θ = 1 - cos² θ.

Let's calculate cos θ and sin θ:

cos² θ = ½ (1 + cos 2θ) = ½ (1 + ⅔) = ½ (⅘) = ⅔

sin² θ = 1 - cos² θ = 1 - ⅔ = ⅘

sin θ = √(⅘) = ⅔ since θ terminates in quadrant 1 where sin is positive

Therefore, the correct answer is sin θ = 4/5 and cos θ = 3/5, which corresponds to option (a).