BASICS OF TRIGONOMETRIC FUNCTIONS
Trigonometric functions in a right-angled triangle are defined for acute angles (less than 90°) and are based on the ratios between the lengths of the sides:
- SINE (sin) is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- COSINE (cos) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- TANGENT (tan) is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
CALCULATING ANGLES:
If we know the lengths of two sides of a right-angled triangle, we can use one of these functions to calculate an unknown angle. The procedure is as follows:
- CHOOSE THE CORRECT TRIGONOMETRIC FUNCTION: First, determine which trigonometric function is appropriate based on the known sides. For example, if we know the length of the hypotenuse and the side opposite the angle, we use sine.
- USE THE INVERSE FUNCTION: To calculate the angle from the ratio, use the inverse trigonometric function (e.g., arcsin, arccos, arctan, also denoted as sin⁻¹, cos⁻¹, tan⁻¹).
- CALCULATE THE ANGLE: Substitute the known ratio into the chosen inverse function to obtain the size of the angle.
EXAMPLE:
Assume we have a right-angled triangle with a hypotenuse of length 10 units and the side opposite angle α has a length of 6 units. Let's calculate the angle α.
- CHOOSE THE TRIGONOMETRIC FUNCTION: Since we know the hypotenuse and the side opposite angle α, we use sine:
sin α = opposite side / hypotenuse = 6 / 10 = 0.6. - USE THE INVERSE FUNCTION:
α = arcsin(0.6) (or α = sin⁻¹(0.6)). - CALCULATE THE ANGLE: Use a calculator or a table of values to calculate arcsin(0.6), which gives an approximate value for the angle α. (Using a calculator, α ≈ 36.87° or ≈ 0.6435 radians).
