Question Describe each of the trigonometric ratios in terms of sides of a right triangle. Trigonometric Ratios in Terms of Sides of a Right Triangle Why can we be sure that sin 30° always has the same value regardless of the size of the right triangle containing the 30° angle?
Trigonometric Ratios in Terms of Sides of a Right Triangle
The right triangle contains three angles, one of which is the right angle itself. The other two are acute angles (Sobecki et al., 2018). The acute angles are where the sides of the triangle meet at its vertex. The trigonometric ratios are defined in terms of these angles and the lengths of the sides of the triangle. The most basic trigonometric ratio is the sine function, which is the ratio of the size of the side opposite to the angle divided by the hypotenuse length. In other words, sin x = opposite ÷ hypotenuse (Sobecki et al., 2018). The cosine function is a ratio between the size of the side adjacent to the angle divided by the hypotenuse length. In other words, cos x equals adjacent length divided by hypotenuse length. Finally, the tangent function is a ratio between the size of the side opposite to the angle divided by the size of the side next to the angle. In other words, tan x = opposite ÷ adjacent. The angle in a right triangle is always measured from the hypotenuse. Therefore, in a right triangle with angle x, sin x = opposite ÷ hypotenuse and cos x = adjacent ÷ hypotenuse.
The value of sin 30° can be found by looking at a right triangle that contains a 30° angle. It always remains the same, no matter how big or small the triangle is (Sobecki et al., 2018). This is because the ratio of the lengths of a right-angled triangle’s sides is always the same, regardless of the triangle’s size. Hence, if one takes any right-angled triangle that contains a 30° angle, they will find that sin 30° = opposite ÷ hypotenuse. In addition, since the hypotenuse is always longer than the opposite side, we can see that sin 30° will always be less than one.
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References
Sobecki, D., Bluman, A. G., Schirck-Matthews, A., & Bluman, A. G. (2018). Math in Our World. McGraw-Hill.