Question

Patrick looked at ΔABC and said cos 60 =x/2. Cindy, who was working with him, said sin 30 =x/2. They asked Ms. Gabrielse who was right. What is the answer she gave them and why? Draw and label the triangle and explain.

Answer 1

They are both right if the triangle is right-angled and the other angles are 60° and 30°, and x=1.

This is because 60+30=90. The adjacent side to the 60° angle is the opposite side to the 30° angle and vice versa. So the sine of one equals the cosine of the other. But if x is not equal to 1, they are both wrong, because sines and cosines are fixed: cos60=sin30=½. That's why they're called trigonometric ratios. If x is a particular length of a side, say, 5cm, then cos60 = x/2 has no meaning, because on one side of the equals sign we have a ratio (a fraction) and on the other side we have a length. If x is just a number but it’s not 1, then neither answer is correct.

In a right triangle where the angles are 60 and 30 degrees, the opposite side to 60° is the adjacent side to 30°, and vice versa. That's why the sine of one and the cosine of the other are equal.