In the triangle, given that cos 𝑥 equals 0.8, what is the value of sin 𝑦?
As we’re dealing with a right-angled triangle, we can use our trigonometrical ratios. This is sometimes referred to as SOHCAHTOA. The S, C, and T stand for the trig functions sine, cosine, and tangent. The O stands for the opposite side to the angle we’re dealing with. The A is the adjacent side. This is next to the angle we’re dealing with. The H stands for the hypotenuse, which is the longest side in any right-angled triangle.
This leads us to three equations. The sin of 𝜃 is equal to the opposite divided by the hypotenuse. The cosine of 𝜃 is equal to the adjacent divided by the hypotenuse. And the tan of 𝜃 is equal to the opposite over the adjacent. The 𝜃 in all three of these formulae is the angle that we’re dealing with. The opposite and the adjacent side will be found according to this angle.
We can begin this question by labeling the three sides of the triangle as 𝑎, 𝑏, and 𝑐. We’re told in the question that cos of 𝑥 or the cos of 𝑥 is equal to 0.8. The cosine of any angle is equal to the adjacent over the hypotenuse. In our case, the side adjacent to angle 𝑥 is letter 𝑎. And the hypotenuse is labeled letter 𝑐. We can therefore say in our triangle that the cos of 𝑥 is equal to 𝑎 over 𝑐.
We need to work out the value of sin of 𝑦. The sin of 𝜃 is equal to the opposite over the hypotenuse. When dealing with angle 𝑦, the hypotenuse is still the longest side of the triangle, labeled 𝑐. The side opposite angle 𝑦 is labeled 𝑎. We can therefore see that the sin of 𝑦 is also equal to 𝑎 over 𝑐. The cos of angle 𝑥 and the sin of angle 𝑦 involve the same two sides, side 𝑎 and side 𝑐. We know that the cosine or cos of 𝑥 is equal to 0.8. This means that the sin of 𝑦 is also equal to 0.8.
We can actually go one stage further here and say that, in any right-angled triangle, the cos of one of the angles 𝑥 will be equal to sine of the other angle 𝑦. If we have a pair of angles that sum to 90 degrees, the cos of one of the angles will be equal to sin of the other angle. For example, cos of 40 degrees is equal to sin of 50 degrees. The cos of 70 degrees is equal to the sin of 20 degrees.
It is worth noting in this case that our value for cos of 𝑥 and sin of 𝑦, 0.8, is equal to four-fifths. This means that the hypotenuse is of length five units and the adjacent to 𝑥 or opposite to 𝑦 is equal to four units. This means that the length of our third side will be three units as we have a Pythagorean triple, a three-four-five triangle. The values of cos of 𝑥 and sin of 𝑦 are both equal to four-fifths or 0.8.