### Video Transcript

Find the value of cos πΌ cos π½ plus sin πΌ sin π½, given sin πΌ is equal to four-fifths, where πΌ is an element of the open interval π by two to π, and five cos π½ minus three equals zero, where π½ is an element of the open interval three π by two to two π.

Weβve been given the value of sin πΌ and an equation involving cos of π½. To answer this question, itβs quite clear weβre going to need to work out the value of cos of πΌ and sin of π. Letβs begin with the information about sin of πΌ. Letβs begin with sin of πΌ. In order to work out the value of cos of πΌ, weβre not going to solve by taking the inverse sin of both sides. Instead, weβre going to recognize that sin of πΌ equals four-fifths can be represented using a right-angle triangle.

We know that the sine ratio tells us that sine of some angle is equal to the length of the opposite side divided by the length of the hypotenuse. So if we construct a right-angle triangle with an included angle of πΌ, the side opposite that must be four units and the hypotenuse must be five. And then, we spot that we have a Pythagorean triple. We know that the numbers three, four, five form a Pythagorean triple. That is, three squared plus four squared equals five squared. And so the side adjacent to our angle πΌ must be three units.

Now, since cos of the angle is equal to the adjacent divided by the hypotenuse, in this exact triangle, we can say that cos of πΌ must be equal to three-fifths. But weβve not yet used this information. That is, πΌ is an element of the open interval from π by two to π radians. In our triangle, weβve assumed that πΌ is an acute angle. So to work out the exact value of cos of πΌ based on the fact that sin of πΌ is equal to four-fifths, weβre going to use the CAST diagram.

Remember, a CAST diagram tells us the sine-tells us the sine of a trigonometric ratio dependent on which quadrant the angle falls in. Our angle is in the open interval from π by two to π radians. So it must lie in this second quadrant. In this quadrant, sine is positive; cos, however, is not. And therefore, it must be negative. So we get that sin of πΌ is equal to four- fifths, but cos of πΌ in this quadrant must be equal to negative three-fifths.

Letβs repeat this process with the information about cos of π½. Weβre going to manipulate the equation a little bit first by adding three to both sides and then dividing through by five. So this time, we find that cos of π½ is equal to three-fifths. Now, in fact, if we go back to our earlier triangle, but this time label the angle π½ instead of πΌ, we know that the adjacent is three and the hypotenuse must be five. So the opposite side in this triangle is still four. So sin of π½ would be equal to four-fifths in this triangle. But weβre going to need to consider the CAST diagram.

This time, our angle is in the open interval from three π by two to two π. This is in the fourth quadrant. And over in this quadrant for angles between three π by two and two π, cos of that angle is positive. This means sine of the angle is negative. And so for cos of π½ is equal to three-fifths, where π½ is an element of the open interval three π by two to two π, sin of π½ is negative four-fifths. And so cos of πΌ times cos of π½ is negative three-fifths times three-fifths. And sin of πΌ times sin of π½ is four-fifths times negative four-fifths.

We multiply fractions by simply multiplying their numerators and then multiplying their denominators. So we get negative nine twenty-fifths plus negative sixteen twenty-fifths. And, of course, since the denominators are equal, we can simply add or subtract their numerators. So we get negative 25 over 25, which is equal to negative one. And so, given the information about sin of πΌ and cos of π½, we find cos of πΌ times cos of π½ plus sin of πΌ times sin of π½ to be equal to negative one.