Given the following figure, find the measures of angles 𝐴𝐶𝐵 and 𝐵𝐴𝐶 and the length of line segment 𝐴𝐶. Give your answers to two decimal places.
We will begin this question by calculating the measure of angle 𝐴𝐶𝐵 using our knowledge of the trigonometric ratios in right triangles. We know that the sin of angle 𝜃 is equal to the opposite over the hypotenuse. The cos of 𝜃 is the adjacent over the hypotenuse. And the tan of angle 𝜃 is equal to the opposite over the adjacent. One way of remembering these three ratios is using the acronym SOHCAHTOA.
Let’s now consider how we can use these to solve the first part of our question. The longest side of a right triangle, which is opposite the right angle, is known as the hypotenuse. The side that is opposite the angle we are working with is the opposite side. Finally, the side that is next to the angle we are working with and the right angle is known as the adjacent. In this question, we have values for the opposite and adjacent. So we will therefore use the tangent ratio. Substituting in our values, we have tan 𝜃 is equal to eight-ninths.
Taking the inverse tangent to both sides gives us 𝜃 is equal to inverse tan of eight-ninths. Ensuring our calculator is in degree mode, we can type this in giving us 𝜃 is equal to 41.6335 and so on. Correct to two decimal places, the measure of angle 𝐴𝐶𝐵 is 41.63 degrees. We could use the tangent ratio once again to calculate the measure of angle 𝐵𝐴𝐶, labeled 𝛼 on our diagram. This time, the opposite and adjacent have swapped places. And the tan of angle 𝛼 is equal to nine over eight. Taking the inverse tangent of both sides, we see that 𝛼 is equal to 48.3664 and so on. The measure of angle 𝐵𝐴𝐶 to two decimal places is 48.37 degrees.
An alternative method here would be to recognize that 𝛼 is equal to 90 degrees minus 𝜃 as the angles in a triangle sum to 180 degrees. This means that we could also have calculated the measure of angle 𝐵𝐴𝐶 by subtracting 41.63 from 90.
The final part of this question asks us to calculate the length of the line segment 𝐴𝐶. This is the hypotenuse of our triangle. And we could calculate this using the trigonometric ratios once again. Alternatively, we can use the Pythagorean theorem, which states that 𝑥 squared plus 𝑦 squared is equal to 𝑧 squared, where 𝑧 is the length of the hypotenuse of the triangle and 𝑥 and 𝑦 are the lengths of the two other sides.
In this question, we have 𝐴𝐵 squared plus 𝐵𝐶 squared is equal to 𝐴𝐶 squared. The left-hand side is equal to eight squared plus nine squared, which is equal to 145. We can then take the square root of both sides, such that 𝐴𝐶 is equal to the square root of 145. Since our answer must be positive, 𝐴𝐶 is equal to 12.0415 and so on. And to two decimal places, this is equal to 12.04 length units. The three answers to this question are 41.63 degrees, 48.37 degrees, and 12.04.