To do trigonometry, start by understanding the three basic trigonometric functions related to a right-angled triangle:
- Sine (sin θ) = Opposite side / Hypotenuse
- Cosine (cos θ) = Adjacent side / Hypotenuse
- Tangent (tan θ) = Opposite side / Adjacent side
You usually label the sides of the triangle relative to the angle θ you are interested in:
- Opposite side is opposite the angle θ,
- Adjacent side is next to the angle θ,
- Hypotenuse is the longest side opposite the right angle.
A common mnemonic to remember these is SOH-CAH-TOA.
Steps to Apply Trigonometry:
- Identify the sides and angle of the right triangle.
- Choose the correct function (sin, cos, or tan) based on which sides or angles you know.
- Apply the formula to find the unknown side or angle.
- For finding angles from ratios, use the inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) on a calculator.
- Use special triangles or the unit circle for exact values for common angles like 30°, 45°, and 60°.
- Use the Pythagorean theorem for missing sides if needed.
For example, to find the opposite side when you know the hypotenuse and angle:
Opposite=sin(θ)×Hypotenuse\text{Opposite}=\sin(\theta)\times \text{Hypotenuse}Opposite=sin(θ)×Hypotenuse
To find an angle when you know two sides:
θ=sin−1(OppositeHypotenuse)\theta =\sin^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)θ=sin−1(HypotenuseOpposite)
This is the foundational process for solving trigonometry problems.
