Answer: In diagram (1), x = 12
Step-by-step explanation: If you observe very closely, there are two triangles placed beside each other, and the one on the left has the right angle on the right side. Notice also that the perpendicular line that separates side 16 from side X, cuts the right angle of the bigger triangle into two, (that is, 90/2 = 45). That means the triangle has three angles, which are 90, 45 and 45 {sum of angles in a triangle equals 180}.
So at this point, what we now have is, an adjacent (side X) and an opposite (side 12). Using trigonometric ratios, (with the angle on the left as the reference angle) what you have is
Tan 45 = Opposite/Adjacent
If you check your calculator or table of values, Tan 45 = 1. Therefore
Tan 45 = Opp/Adj
1 = 12/X
By cross multiplication we now have
X = 12
For the diagram on the right side, Cos B can be derived as
Adjacent/Hypotenuse
However if we use a as the reference angle, then it becomes
Sin a = Opposite/Hypotenuse
Note however that when using “a” as the reference angle, the opposite side becomes the adjacent of angle “B.” This means that, the sides that would result in Cos B are the same sides that would result in Sin a. Therefore Cos B would be equal in value to Sin a. Hence Sin a is also 0.68.
For the second triangle in the right side diagram, Cos A would be derived as
Cos A = Adjacent/Hypotenuse.
However in that same diagram, the adjacent of angle A would be the opposite when using angle B as the reference angle. In other words we can write
Sin B = Opposite/Hypotenuse
Knowing that the adjacent side of A is the same as the opposite side of B
The correct answer is Cos A = Sin B
