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T 2 Right Triangle Trigonometry
Trigonometry is a branch of mathematics involving the study of tri- angles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. The ability to use and manipulate trigonometric functions is necessary in other branches of mathemat- ics, 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).
Trigonometric Ratios
In a right-angled triangle the following ratios are defined for a given angle θ
sine θ = (opposite side length) (hypotenuse length) cosine θ = (adjacent side length) (hypotenuse length)
tangent θ = (opposite side length) (adjacent side length)
These ratios are abbreviated to sin θ , cos θ , and tan θ respectively.
A useful memory aid is SOH CAH TOA:
S in θ = O H ppyp Cos θ = (^) HA djyp T an θ = O A ppdj
These ratios can be used to find unknown sides and angles in right-angled triangles.
Examples
Evaluating ratios
In the right-angled triangle below evaluate sin θ , cos θ , and tan θ.
Finding angles
Find the value of the angle in the triangle below^1 1 Step 1 : Determine which ratio to use Step 2 : Write the relevant equation Step 3 : Substitute the values Step 4 : Solve the equation
The ratio that relates these two sides is the tangent ratio.
tan θ = Opp Adj tan 27 °^ = b 42 (tan 27 °) × 42 = b b = 21.4cm
(b)
In this problem we know an angle and the adjacent side. The side to be determined is the hypotenuse. The ratio that relates these two sides is the cosine ratio.
cos θ = Adj Hyp cos 35 °^ =
x (cos 35 °) × x = 7 x =
cos 35 ° x = 8.
Special angles and exact values
There are some special angles for which the trigonometric functions have exact values rather than decimal approximations. Applying the rules for sine, cosine and tangent to the triangles below, exact values for the sine, cosine and tangent of the angles 30°, 45 °^ and 60°^ can be found.
s in 45 °^ = √^1 2
cos 45 °^ = √^1 2
tan 45 °^ = 1
s in 60 °^ =
cos 60 °^ =
tan 60 °^ =
s in 30 °^ =
cos 30 °^ =
tan 30 °^ =
Exercises
Exercise 1
Using the right-angled triangle below find: (a) sin θ , (b) tan θ , (c) cos α , (d) tan α (Hint: Use Pythagoras theorem to find the hypotenuse)
Answers
Exercise 1 (a) 1213 = 0. 9231 (b) 125 = 2. 4 (c) 1213 = 0. 9231 (d) 125 =
- 4167 Exercise 2 (a) = 6. 6 cm (b) a = 3. 4 mm (c) z = 7. 8 cm, α = 37. 7 ◦^ (d) θ = 18. 8 °, x = 19. 1 (e) a = 44. 4 (f) b = 47. 1 (g) 11. 6 mm. and 17. 5 mm.
