Video: Solving Trigonometric Equations by Squaring

Video Transcript

By squaring both sides or otherwise, solve the equation. Four sin 𝜃 minus four cos 𝜃 is equal to root three, where 𝜃 is greater than zero but less than or equal to 360 degrees. Be careful to remove any extraneous solutions. Give your answers to two decimal places.

Whilst the question suggests squaring both sides, it will be easier to use one of our trigonometrical identities. One of our addition formulae states that sin of 𝜃 minus 𝛼 is equal to sin 𝜃 multiplied by cos 𝛼 minus cos 𝜃 multiplied by sin 𝛼. Using this identity allows us to write any equation in the form 𝑎 sin 𝜃 minus 𝑏 cos 𝜃 in the form 𝑟 multiplied by sin of 𝜃 minus 𝛼. The value of 𝑟 is equal to the square root of 𝑎 squared plus 𝑏 squared, where 𝑎 and 𝑏 are the coefficients of sin 𝜃 and cos 𝜃. And the angle 𝛼 is equal to the inverse tan of 𝑏 over 𝑎.

In our equation four sin 𝜃 minus four cos 𝜃, then 𝑎 is equal to four and 𝑏 is also equal to four. This means that 𝑟 is equal to the square root of four squared plus four squared. Four squared is equal to 16. Therefore, 𝑟 is equal to the square root of 16 plus 16. This simplifies to 𝑟 is equal to root 32. Root 32 can be rewritten as root 16 multiplied by root two. As the square root of 16 is equal to four, 𝑟 is equal to four root two. The angle 𝛼 is equal to inverse tan or tan to the minus one of four divided by four. Four divided by four is equal to one. The inverse tan of one is equal to 45 degrees, as tan of 45 equals one.

This means that the expression four sin 𝜃 minus four cos 𝜃 can be rewritten as four root two sin of 𝜃 minus 45. As four sin 𝜃 minus four cos 𝜃 was equal to root three, four root two sin 𝜃 minus 45 will also be equal to root three. In order to calculate our values for 𝜃, we firstly need to divide both sides of the equation by four root two. On the left-hand side, this leaves us with sin of 𝜃 minus 45. On the right-hand side, we’re left with root six over eight or 0.3061 as a decimal.

Our next step is to take the inverse of sin or sin to the minus one of both sides. This gives us 𝜃 minus 45 is equal to sin to the minus one of 0.3061, etcetera. Typing this into the calculator gives us an answer of 17.83 to 2 decimal places. Adding 45 to this will give us one of our solutions. However, we want all the solutions between zero and 360 degrees. One way of doing this is by considering the sin graph between zero and 360 degrees. We know that this is wave-shaped and is symmetrical. We can see from the graph that the sin of 17.83 is equal to the sin of 162.17, as they’re both at the same height on the 𝑦-axis.

This means that the inverse sin of 0.3061, etcetera, is also equal to 162.17. An alternative method to find the second solution would be to remember one of our rules. This is that the second solution will be 180 degrees minus the first solution. This is also sometimes known as the cast method. If the sin of 𝜃 is equal to a positive value, there’ll be one solution between zero and 90 degrees and another solution between 90 on 180 degrees. Our final step in this question is to add 45 to both of our solutions. 17.83 plus 45 is equal to 62.83 and 162.17 plus 45 is equal to 207.17. Our two solutions for 𝜃 are 62.83 degrees and 207.17 degrees. Both of these have been given to two decimal places.