Question Video: Solving Trigonometric Equations Involving Half Angles

Solve tan (𝑥/2) = sin 𝑥, where 0 ≤ 𝑥 < 2𝜋.

02:46

Video Transcript

Solve tan of 𝑥 over two is equal to sin 𝑥, where 𝑥 is greater than or equal to zero and less than two 𝜋 radians.

We are told in the question that our solutions must be less than two 𝜋 and greater than or equal to zero. This means that we will be able to solve the equations using the CAST diagram or by sketching our trig graphs. Using one of our half-angle identities, we know that tan of 𝑥 over two is equal to one minus cos 𝑥 divided by sin 𝑥. This means that we can rewrite our equation as one minus cos 𝑥 divided by sin 𝑥 is equal to sin 𝑥

Multiplying both sides of the equation by sin 𝑥 gives us one minus cos 𝑥 is equal to sin squared 𝑥. As sin squared 𝑥 is equal to one minus cos squared 𝑥, we have one minus cos 𝑥 is equal to one minus cos squared 𝑥. We can then add cos squared 𝑥 and subtract one from both sides of our equation. This gives us cos squared 𝑥 minus cos 𝑥 is equal to zero. Factoring out cos 𝑥 gives us cos 𝑥 multiplied by cos 𝑥 minus one is equal to zero. This means that either cos of 𝑥 equals zero or cos of 𝑥 minus one equals zero. The second equation can be rewritten as the cos of 𝑥 equals one.

We now have two equations that can be solved either by using the CAST diagram or by drawing the graph of cos 𝑥. The cosine graph between zero and two 𝜋 looks as shown. Remembering that our values need to be greater than or equal to zero and less than two 𝜋, we see that the graph is equal to zero when 𝑥 is equal to 𝜋 over two and three 𝜋 over two. cos of 𝑥 is equal to one when 𝑥 is equal to zero or two 𝜋. However, two 𝜋 is not within the range of values for 𝑥. This means that there are three solutions to our equation. The tan of 𝑥 over two is equal to the sin of 𝑥 when 𝑥 is equal to zero, 𝜋 over two, and three 𝜋 over two.

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