Trigonometry is a branch of mathematics involving the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. Also the ability to use and manipulate trigonometric functions is necessary in other branches of mathematics, including calculus, vectors and complex numbers. Right-angled Triangles In a right-angled triangle the three sides are given special names. The side opposite the right angle is called the hypotenuse (h) – this is always the longest side of the triangle. The other two sides are named in relation to another known angle (or an unknown angle under consideration).
If this angle is known or under consideration
this side is called the opposite side because it is opposite the angle
This side is called the adjacent side because it is adjacent to or near the angle Trigonometric Ratios In a right-angled triangle the following ratios are defined sin θ = opposite side length o = hypotenuse length h cosineθ = adjacent side length a = hypotenuse length h
opposite side length o = adjacent side length a
where θ is the angle as shown
These ratios are abbreviated to sinθ, cosθ, and tanθ respectively. A useful memory aid is Soh Cah Toa pronounced ‘so-car-tow-a’
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Unknown sides and angles in right angled triangles can be found using these ratios. Examples Find the value of the indicated unknown (side length or angle) in each of the following diagrams. (1) Method 1. Determine which ratio to use. 2. Write the relevant equation. 3. Substitute values from given information. 4. Solve the equation for the unknown.
In this problem we have an angle, the opposite side and the adjacent side. The ratio that relates these two sides is the tangent ratio. tan θ = opposite side adjacent side
Substitute in the equation: (opposite side = b, adjacent side = 42, and θ = 27o)
b 42 b = 42 × tan 27° b = 21.4 tan 27° =
transpose to give
In this triangle we know two sides and need to find the angle θ. 13.4 cm The known sides are the opposite side and the hypotenuse. The ratio that relates the opposite side and the hypotenuse is the sine ratio.
opposite side hypotenuse 13.4 sin θ = 19.7 sin θ = 0.6082 sin θ =
opposite side = 13.4cm. hypotenuse = 19.7cm.
This means we need the angle whose sine is 0.6082, or sin −1 0.6082 from the calculator. ∴ θ = 42.90
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Pythagoras’ Thoerem Pythagoras’ Theorem states that in a right angled triangle the square of the length of the hypotenuse side (h), is equal to the sum of the squares of the other two sides.
Pythagoras’ Theorem can be used to find a side length of a right angled triangle given the other two side lengths
find the value of h h =6 +8 ∴ h 2 = 100 ∴ h = 10
2 2 2
∴ h 2 = 36 + 64
6 cm h
Pythagoras’ Theorem for this triangle
square root of 100
Note Measurements must be in the same units and the unknown length will be in these same units so h will be 10 cm
find the value of x
4.2 2 = 2.7 2 + x 2
∴17.64 = 7.29 + x 2 ∴10.35 = x 2 ∴ x = 3.22
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Exercise Find the value of the indicated unknown (side length or angle) in each of the following diagrams.
4.71 mm a 62o a 14 cm
(c) 4.8 cm
6.2 cm 20.2
b 34 27o 42
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Special angles and exact values
There are some special angles that enable us to obtain exact solutions for the functions and tan. sin, cos
If we take the two triangles below, and apply the basic trigonometry rules for sine, cosine and tangent –
1 From these two triangles, exact answers for sine,...