Concept Check Does there exist an angle θ with the function value...

Question

Concept Check Does there exist an angle θ with the function values cos θ = ⅔ and sin θ = ¾?

Accepted Answer

Hello. Today we're going to be determining if there is an angle theta that allows the cosine function to equal to five divided by six and the sine function to equal to one divided by the square root of three. So before we approach this problem, let's just go ahead and write down some of the given information, we are told that we want an angle data that allows the cosine function to equal to five divided by six. So we can say that cosine of data will equal to five divided by six. We want also want this angle theta to allow the sine function to equal to equal to one divided by the square root of three. So we can say that sine of data is equal to one divided by the square root of three. Now, in order to determine whether there is an angle theta that allows these two functions to equal to these values. At the same time, we're going to need the help of a Pythagorean theorem to prove this. Now we have a Pythagorean theorem that states that cosine squared data plus sine squared data is equal to one. So what this means is that if we plug in the corresponding cosine and sine values into this identity and it equals to one, then that will prove that there is an angle theta that allows the cosine and sine function to equal to the corresponding values. So we're going to plug in cosine data equal to five divided by six into the identity. But notice that the cosine function is squared on the left hand side of the identity. That means when we plug in the value of five divided by six, we're going to need to square the value. So we have five divided by six squared. And the same can be said for the sine function, the sine function is squared. So when we plug in the value of one divided by the square root of three, we're going to have to square that value. So we can rewrite the identity as five divided by six squared plus one divided by the square root of three squared. And that should equal to one. Now, five divided by six squared will simplify to leave us with 25 divided by 30 one divided by the square root of three squared will leave us with the value of one divided by three. Now, what we need to do is we need to add the two fractions together. In order to add the two fractions together, we need to have the same denominators for one divided by the square root of three. If we multiply the numerator and denominator by 10, we can get a denominator of 30. So one divided by the square, one divided by three, multiplied by 10, divided by 10 will leave us with the value of 10 divided by 30. So we have 25 divided by 30 plus 10, divided by 30. And that should equal to one 25 divided by plus 10, divided by 30 will give us the value of 35 divided by 30. We can simplify this fraction by dividing the numerator and denominator by five. That will leave us with seven divided by six equal to one. Now seven divided by six is not equal to one. This is a false statement. So what this means is that there is not an angle theta that allows the cosine and sine function to equal to five divided by six and one divided by the square root of three. And with that being said, the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Concept Check Does there exist an angle θ with the function values cos θ = ⅔ and sin θ = ¾?

Key Concepts

Pythagorean Identity

Range of Trigonometric Functions

Existence of Angles in Trigonometry

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