Understanding Trigonometric Ratios
In a right-angled triangle, the trigonometric ratios are defined as follows:
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Sine (sin) of an acute angle is the ratio of the length of the opposite side to the hypotenuse.
- Formula: ( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} )
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Cosine (cos) of an acute angle is the ratio of the length of the adjacent side to the hypotenuse.
- Formula: ( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} )
-
Tangent (tan) of an acute angle is the ratio of the length of the opposite side to the adjacent side.
- Formula: ( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} )
To remember these relationships, use the acronym SOH-CAH-TOA:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
These ratios are crucial for solving problems involving right-angled triangles, especially in geometry and physics.
Key points to remember
- Sine = Opposite side / Hypotenuse
- Cosine = Adjacent side / Hypotenuse
- Tangent = Opposite side / Adjacent
- Use SOH-CAH-TOA to remember definitions
- Apply ratios in right-angled triangle problems
Worked example
Define sine, cosine, and tangent for angle A in triangle ABC where AB = 3, AC = 4, and BC = 5.
- ( \sin(A) = \frac{3}{5} )
- ( \cos(A) = \frac{4}{5} )
- ( \tan(A) = \frac{3}{4} )