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Question Video: Solving Trigonometric Equations Mathematics • 10th Grade

Find the measure of ∠𝐴 given 17 sin 𝐴 − 16 = 0, where 𝐴 ∈ (0, 𝜋/2). Give your answer to the nearest second.


Video Transcript

Find the measure of angle 𝐴 given 17 sin 𝐴 minus 16 is equal to zero, where 𝐴 is a value between zero and 𝜋 by two. Give your answer to the nearest second.

We immediately notice that we are asked to give our answer to the nearest second, whereas we are told that 𝐴 is an angle that is between zero and 𝜋 by two radians. Using a CAST diagram, we see that we need to calculate the angle in the first quadrant. We know that the angle 𝜋 by two radians is the same as 90 degrees. Therefore, we need to calculate the solution between zero and 90 degrees. We will begin by adding 16 to both sides of our equation such that 17 sin 𝐴 is equal to 16. We will then divide both sides of this equation by 17 such that the sin of angle 𝐴 is equal to 16 over 17 or sixteen seventeenths.

Our next step is to take the inverse sin of both sides of this equation. Angle 𝐴 is equal to the inverse sin of 16 over 17. Ensuring our calculator is in degree mode, we type the right-hand side into the calculator, giving us an answer of 70.2500 and so on. As we need to give our answer to the nearest second, we will need to convert the decimal part of this answer to minutes and then seconds. We can do this by pressing the degrees, minutes, and seconds button on a scientific calculator. This gives us an answer of 70 degrees, 15 minutes, and zero seconds. The angle 𝐴 is therefore equal to 70 degrees and 15 minutes to the nearest second.

An alternative method to convert to minutes and seconds would be to multiply the decimal part of our answer by 60. 0.2500 and so on multiplied by 60 is equal to 15.0042 and so on. This confirms that we have 15 minutes. And we can then multiply the decimal answer of this number by 60 to calculate the seconds. This is equal to 0.252 and so on. As this would round down to zero to the nearest whole number, we have zero seconds. This confirms that the correct answer is 70 degrees, 15 minutes, and zero seconds.

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