In Exercises 21–24, θ is an acute angle and sin θ is given. Use t...

Question

In Exercises 21–24, θ is an acute angle and sin θ is given. Use the Pythagorean identity sin²θ + cos²θ = 1 to find cos θ.__sin θ = √398

Accepted Answer

Hello everybody night today. Today, we're going to look again at this question that states the trigonometric function sine of alpha is given. And they're asking us to find the value of cosine of alpha by using an appropriate Python and identity. And we should note that alpha is an acute angle. Now, they tell us that sign of alpha is equal to open parentheses, screwed of close parentheses divided by nine. Now, the others provided are a or divided by nine B square root of five, divided by nine C five, divided by nine and D square root of two divided by nine. Now, the first thing we have to see is the Python and identity. So let's recall that when it comes to Python Pythagorean identity, this states that sine squared X plus cosine squared X is equal to one. So how will this help us in our question? Well, we can do some man manipulation. So we will therefore have that cosine squared X is equal to one minus sine squared X. And all we did there was move the sine squared X from the left hand side to the right hand side by subtracting So if we want to solve for cosine of X, we just take the square root of both sides. And we'll have that cosine of X is equal to the square root of one minus sine squared X. And if we compare this to our question, we'll have cosine of alpha. It's equal to the square root of one minus sine squared alpha. And now we can start doing some plugging in. So we'll have coin of alpha is equal to the square root of one minus open parentheses, square root of 65 divided by nine closed parentheses squared. And now we can do the squaring and we'll have cosine of alpha is equal to the square root of one minus. So if we have a fraction squared, we have to square both the numerator and the denominator. So the square root of 65 squared would just be 65 and we'll have that divided by nine squared, which is 81. So inside of our square root, we'll have one minus 65 divided by 81. And we can continue and we'll have coin of alpha is equal to the square root of, and that will turn our one into a fraction. So if we have our other fraction is divided by 81 that means one will be 81 divided by 81. And we'll have that minus 65 divided by 81. And now we can subtract and we'll have cosine of alpha is equal to the square root of 16 divided by 81. And when we have a fraction inside of a square root, we can distribute that square root to both the numerator and the denominator. So have coin of alpha is equal to the square root of 16 divided by the square root of 81. And we know the square root of 16 and the squared of 6 81. So we have that cosine of alpha is equal to the squared of 16 is four. So we have four divided by the squared of 81 which is nine. So we have that cosine of alpha is equal to four divided by nine or answer choices. A. So as you can see, we can turn this question that might look a little bit complicated and just using a simple Python identity, we were able to solve it step by step. So I hope that was helpful. And until next time.

In Exercises 21–24, θ is an acute angle and sin θ is given. Use the Pythagorean identity sin²θ + cos²θ = 1 to find cos θ.__sin θ = √398

Key Concepts

Pythagorean Identity

Sine Function

Cosine Function

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