Find the value of sin 𝛼 cos 𝛽 minus cos 𝛼 sin 𝛽, given tan 𝛼 is equal to three-quarters, where 𝛼 is the smallest positive angle, and tan 𝛽 is equal to fifteen eighths, where 𝛽 is greater than 180 degrees and less than 270 degrees.
We begin by using our knowledge of the CAST diagram to identify which quadrant angles 𝛼 and 𝛽 lie in. Since tan 𝛼 is positive and 𝛼 is the smallest positive angle, from our CAST diagram, we see that 𝛼 must lie in the first quadrant and is therefore between zero and 90 degrees. We are told that 𝛽 is greater than 180 degrees and less than 270 degrees. So it therefore lies in the third quadrant.
As tan 𝛼 is equal to three-quarters, we can sketch a right triangle in the first quadrant as shown, recalling that the tangent of any angle in a right triangle is equal to the opposite over the adjacent. The triangle is a Pythagorean triple consisting of three positive integers — three, four, and five — such that three squared plus four squared is equal to five squared. We know that the sin of angle 𝜃 is equal to the opposite over the hypotenuse and the cos of angle 𝜃 is equal to the adjacent over the hypotenuse. In the first quadrant, the sine, cosine, and tangent of any angle are all positive. From the diagram, we can therefore see that sin 𝛼 is equal to three-fifths and cos 𝛼 is four-fifths.
We will now repeat this process using the fact that tan 𝛽 is equal to fifteen eighths. This time, we sketch a right triangle in the third quadrant, where we know the tangent of any angle is positive and the sine and cosine of any angle are negative. We have another Pythagorean triple consisting of the three positive integers eight, 15, and 17 such that eight squared plus 15 squared is equal to 17 squared. Since angle 𝛽 must be measured from the positive 𝑥-axis, we will call the angle between the negative 𝑥-axis and the hypotenuse of our triangle 𝛾. Using the sine and cosine trigonometric ratios, we have sin 𝛾 is equal to fifteen seventeenths and cos 𝛾 is equal to eight seventeenths. We can see from our diagram that 𝛽 is equal to 180 degrees plus 𝛾.
Recalling our related angle properties, we know that the sin of 180 degrees plus 𝜃 is equal to negative sin 𝜃. This means that the sin of 180 degrees plus 𝛾 is equal to negative sin 𝛾. sin 𝛽 is therefore equal to negative sin 𝛾, which is equal to negative fifteen seventeenths. In the same way for the cosine function, the cos of 180 degrees plus 𝛾 is equal to negative cos 𝛾. cos 𝛽 is therefore equal to negative cos 𝛾, which is equal to negative eight seventeenths. At this point, it is worth recalling that our values of sin 𝛽 and cos 𝛽 must both be negative as angle 𝛽 lies in the third quadrant.
We are now in a position to substitute our values of sin 𝛼, cos 𝛼, sin 𝛽, and cos 𝛽 into our expression. This is equal to three-fifths multiplied by negative eight seventeenths minus four-fifths multiplied by negative fifteen seventeenths. Simplifying this, we have negative 24 over 85 minus negative 60 over 85. Since the denominators are the same, we can simply add 60 to negative 24, giving us 36 over 85. The value of sin 𝛼 cos 𝛽 minus cos 𝛼 sin 𝛽 is 36 over 85.