Solving trigonometric equations Solve the following equations....

Question

Solving trigonometric equations Solve the following equations.

cos²Θ = 1/2 , 0 ≤ Θ < 2π

Accepted Answer

Hello. Today we're going to be solving for the angle A in the following trigonometric equation, and we're going to be writing our answers in terms of pi. So the trigonometric equation given to us is cosine squared of A equal to 1 divided by 4, and this is where the angle A lies within the first rotation of the unit circle between 0 and 2 pi. So how can we go about solving for a in the given equation? Well notice that on the left hand side of the equation, cosine is a squared quantity. We can simplify this by taking the square root of both sides of the equation. Doing so will allow us to simplify the left hand side to cosine of a and on the right hand side, the square root of 1 divided by 4 will simplify to 1 divided by 2. But we need to be careful because whenever you take the square root of a number, we end up with 2 solutions. A positive and negative solution. So what we are left with is cosine of A is equal to +onedwo. We can split this equation into 2 separate equations. The first equation we get is cos is equal to positive onetwo and the second equation is cosine of A is equal to negative one divided by 2. In order to answer these questions, we're going to need to use a unit circle. So let's go ahead and draw that very quick. Now, let's take a look at the left hand equation. We need to ask ourselves what angles of a can we plug into cosine that will give us the value of positive 1 divided by 2? Now if you're unsure of how to answer this, recall that every terminal point on the unit circle is defined by cosine comma sine. This is where the cosine value takes the x position of any terminal point on the unit circle. So what this means is what angles on the unit circle have the value of positive 1 divided by 2 in the x position of the terminal point? Well, the first angle is located at pi divided by 3. At this angle, we have the terminal point 1 divided by 2 comma square root 3 divided by 2 and the second angle where cosine of a is equal to a half will be located at 5pithree. At this angle, the terminal point is given to us as 1 divided by 2 comma negative square root 3 divided by 2. So what this means is that we have 2 solutions for a in the first equation. A is equal to pi divided by 3 and 5pithree. Now, we're going to take a look at the second equation. Now, just like before, we need to ask ourselves, what angles of a can we plug into cosine that will give us the value of negative one divided by 2? Or in other words, what angles on the unit circle have the value negative one divided by 2 in the x position of the terminal point? Well, the first angle will be located at 2pithree. At this angle, we have the terminal point negative one divided by 2 comma square root 3 divided by 2 and the second angle will be located at 4pithree. At this angle, we have the terminal point negative one divided by 2 comma negative square root 3 divided by 2. So what this means is that there are 2 solutions for the second equation. A is equal to 2pithree and 4pithree. Now if we combine all the solutions of A together, we have 4 total solutions that satisfy the original trigonometric equation. A is equal to pi divided by 3, 2 pi divided by 3, 4 pi divided by 3, and 5 pi divided by 3, and these four angles are going to be the solutions of a that satisfy the original equation. 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.

Solving trigonometric equations Solve the following equations.

cos²Θ = 1/2 , 0 ≤ Θ < 2π

Key Concepts

Trigonometric Functions

Inverse Trigonometric Functions

Periodic Nature of Trigonometric Functions

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