Trigonometric Functions – Example

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:

1. 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.
2. 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).
3. 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.

1. 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.
2. USE THE INVERSE FUNCTION:
   α = arcsin(0.6) (or α = sin⁻¹(0.6)).
3. 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).