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Trigonometry in right-angled triangles
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Similar right-angled triangles share the same side ratios. These ratios, sine, cosine and tangent, link the angles to the sides.
Naming the sides
Hypotenuse (H): opposite the right angle, the longest side.
Opposite (O): opposite the angle you are using.
Adjacent (A): next to that angle (but not the hypotenuse).
Opposite (O): opposite the angle you are using.
Adjacent (A): next to that angle (but not the hypotenuse).
SOH CAH TOA
sin θ = OH cos θ = AH tan θ = OA
These ratios are the same in every similar right-angled triangle, because the shapes are enlargements of each other.
These ratios are the same in every similar right-angled triangle, because the shapes are enlargements of each other.
Using the ratios
Finding a side: opposite = 10 × sin 30° = 5.
Finding an angle: use the inverse button. If sin θ = 59, θ = sin−1(59) = 33.7°.
Check your calculator is in degrees.
Finding an angle: use the inverse button. If sin θ = 59, θ = sin−1(59) = 33.7°.
Check your calculator is in degrees.
A right-angled triangle has hypotenuse 12 cm and an angle of 40°. Find the side adjacent to the 40° angle, to 1 decimal place.
- You know H and want A: use cos
- A = 12 × cos 40°
- = 9.19...
Answer: 9.2 cm
Label the sides O, A and H from the angle, pick the ratio that uses the two sides you know or want (SOH CAH TOA), and use inverse functions to find angles.
The interactive lesson includes the diagrams for this topic.
Check you have got it
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