Video: Solving Trigonometric Equations Using the Addition Formulas or by Squaring

Solve √(2) sin 𝜃 +√(3) cos 𝜃 = 2, where 0 < 𝜃 ≤ 2𝜋. Give your answer in radians to three significant figures.

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Video Transcript

Solve root two sin 𝜃 plus root three cos 𝜃 is equal to two, where 𝜃 is greater than zero but less than or equal to two 𝜋 radians. Give you answer in radians to three significant figures.

In order to answer this question, we’ll use one of the addition formulas from trigonometry. This states that sin of 𝜃 plus 𝛼 is equal to sin 𝜃 cos 𝛼 plus cos 𝜃 sin 𝛼. This allows us to rewrite any expression of the form 𝑎 sin 𝜃 𝑏 cos 𝜃 as 𝑅 of sin 𝜃 plus 𝛼. We can calculate the values of 𝑅 and 𝛼 using the coefficients 𝑎 and 𝑏. 𝑅 is equal to the square root of 𝑎 squared plus 𝑏 squared. And the angle 𝛼 is equal to tan to the minus one, or the inverse tan, of 𝑏 over 𝑎.

Our expression in this question is root two sin 𝜃 plus root three cos 𝜃. Our value for 𝑎 is root two. And our value for 𝑏 is root three. This means that 𝑅 is equal to the square root of root two squared plus root three squared. Root two squared is equal to two. And root three squared is equal to three. As two plus three is equal to five, 𝑅 is equal to root five. The angle 𝛼 is equal to the inverse tan of root three over root two. Ensuring that our calculator is in radian mode, typing this in gives us an answer of 0.886 and so on. It is important that we do not round this answer at this point.

The expression root two sin 𝜃 plus root three cos 𝜃 can, therefore, be rewritten as root five sin of 𝜃 plus 0.886 etcetera. We want to solve this expression equal to two. Our first step to solve the equation is to divide both sides by root five. On the left-hand side, the root fives cancel, so we are left with sin of 𝜃 plus 0.886 and so on. On the right-hand side, we have two divided by root five.

Our next step is to take the inverse sin, or sin to the minus one, of both sides of the equation. The left-hand side becomes 𝜃 plus 0.886 and so on. On the right-hand side, we have sin to the minus one of two divided by root five. This is equal to 1.107 and so on. However, once again, it is important to not round the answer at this stage. One solution can, therefore, be found by subtracting 0.886 from 1.107. However, we have been asked to calculate all the solutions between zero and two 𝜋.

One way of calculating the other solutions would be to draw the sin graph. Alternatively, we can use the cast method. The cast method tells us that if the sin of an angle is equal to a positive number, there are a two solutions between zero and two 𝜋. One of the solutions will lie between zero and 𝜋 by two. And the second solution will be between 𝜋 by two and 𝜋. As the sin graph is symmetrical, if one solution is 1.107, then the second solution will be 𝜋 minus this answer. This is equal to 2.34 and so on.

𝜃 plus 0.886 and so on is equal to 1.107 etcetera, or 2.034 etcetera. We need to subtract 0.886 and so on from both sides of the equation. Subtracting 0.886 from 1.107 gives us 0.221. And subtracting 0.886 from 2.34 gives us 1.148. As we were asked to give our answers to three significant figures, this is equal to 1.15. The solutions to the equation root two sin 𝜃 plus root three cos 𝜃 equals two between zero and two 𝜋 are 0.221 and 1.15.

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