Trigonometry
Trigonometric ratios, identities and equations
Trigonometric Ratios
In a right-angled triangle, the three basic trigonometric ratios are:
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
Remember: SOH-CAH-TOA.
Example
Finding a Side
A right triangle has hypotenuse = 10 cm and angle θ = 30°. Find the opposite side.
sin(30°) = opposite / 10
0.5 = opposite / 10
opposite = 10 × 0.5 = 5 cm
Example
Finding an Angle
A right triangle has opposite = 4 cm and adjacent = 4 cm. Find angle θ.
tan(θ) = 4/4 = 1
θ = tan⁻¹(1) = 45°
Special Angles
Memorise these values:
sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3
sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1
sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3
Note
Remember
SOH-CAH-TOA only works in right-angled triangles. The hypotenuse is always the longest side, opposite the right angle. Label your triangle clearly before choosing which ratio to use.
Key Vocabulary
TrigonometryThe study of relationships between angles and sides of triangles
HypotenuseThe longest side of a right triangle, opposite the 90° angle
OppositeThe side across from the angle being considered
AdjacentThe side next to the angle (not the hypotenuse)
SineRatio of opposite to hypotenuse: sin(θ) = opp/hyp
CosineRatio of adjacent to hypotenuse: cos(θ) = adj/hyp
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Trigonometry
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