If θ is an acute angle and sin θ = (2√7) / 7, use the identity...

Question

If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

Accepted Answer

Hello, everyone. We are going to find the value of the cosine of beta. If the sign of beta equals three radical 13 divided by 13 and beta is an acute angle, we have four answer choices. We are gonna use the Pythagorean identity that states that sine squared beta plus cosine squared beta will equal one. So plugging in what we know for sign, we can say we have three radical 13 divided by squared plus the cosine squared beta equals one squaring three radical 13 divided by 13. When I square, the top I will have nine multiplied by 13 and that'll be divided by 13 squared plus cosine squared beta equals one. I left the denominator of that fraction as 13 squared because I know I can reduce the 13 in the numerator with one of the thirteens in the denominator. So now I have 9/13 plus cosine squared beta equals one subtracting 9/13 from both sides, I get the cosine squared of beta equals 4/13. The opposite of a square is a square root. So I will take the square root of 4/13. And this will tell me that the cosine of beta is plus or minus two divided by radical 13. Since beta is an acute angle, we are gonna have a positive answer. So we're sticking with the positives. I'm going to rationalize my denominator by multiplying the numerator and denominator by radical 13. And this will give me the value to radical 13 divided by 13, which is our value for the cosine of beta and is answer choice A have a nice day.

If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

Key Concepts

Pythagorean Identity

Sine Function and Its Range

Sign of Cosine in the First Quadrant

Watch next