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Trigonometry
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- 2. Table of contents 1.2. 3. 4. 5. Basic trig SINE RULE COSINE RULE (Side) COSINE RULE (Angle) QUESTIONS
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- 4. RIGHT ANGLED TRIANGLES theside opposite to the right angle is called the hypotenuse
- 5. RIGHT ANGLED TRIANGLES hypotenuse theside that is close to or touching the angle is called the adjacent the side that is opposite the angle is called the opposite θ
- 6. hypotenuse adjacent opposite Sine is theratio of the opposite to the hypotenuse 𝑠𝑖𝑛𝑒(𝜃) = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Cosine is the ratio of the adjacent to the hypotenuse 𝑐𝑜𝑠𝑖𝑛𝑒(𝜃) = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 Tangent is the ratio of the opposite to the adjacent 𝑡𝑎𝑛𝑔𝑒𝑛𝑡(𝜃) = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 RIGHT ANGLED TRIANGLES θ
- 7. H A O A common mnemonicused to remember these rules is SOH CAH TOA 𝑺 = 𝑶 𝑯 𝑪 = 𝑨 𝑯 𝑻 = 𝑶 𝑨 RIGHT ANGLED TRIANGLES
- 8. 4 cm hypotenuse x adjacent 6 cm Tofind x: The hypotenuse is given and the adjacent is missing so, cosine is used 𝑐𝑜𝑠𝑖𝑛𝑒(𝜃) = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 𝑐𝑜𝑠 32 = 𝑥 4 𝑐𝑜𝑠 32 4 = 𝑥 x = 3.34 cm EXAMPLE 32̊ opposite ·
- 9. Sine inverse isused when both the opposite and the hypotenuse are given BUT the angle is missing Cosine inverse is used when both the adjacent and the hypotenuse are given BUT the angle is missing Tangent inverse is used when both the opposite and the adjacent are given BUT the angle is missing INVERSES s𝑖𝑛𝑒−1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝜃 cos𝑖𝑛𝑒−1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝜃 2 𝑡𝑎𝑛𝑔𝑒𝑛𝑡−1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 = 𝜃 SO THE INVERSES ARE USED TO FIND THE ANGLE
- 10. 4 cm hypotenuse 3.34 cm adjacent 6cm To find x: The hypotenuse is given and the adjacent is given so, cosine inverse is used cos𝑖𝑛𝑒−1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 𝑎𝑛𝑔𝑙𝑒 cos−1 3.34 4 = 𝑥 x = 33.4º EXAMPLE xº opposite
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- 12. Sine Rule The sinerule enables you to calculate lengths and angles in any triangle There are two versions of this rule (one for calculating the side and one for the angle)
- 13. Sine Rule The sinerule is used to find: a side when two angles and an opposite side are known an angle when two sides and an opposite angle are known.
- 14. A B C b a c Sine Rulefor side 𝑎 𝑠𝑖𝑛 𝐴 = 𝑏 𝑠𝑖𝑛𝐵 = 𝑐 𝑠𝑖𝑛𝐶
- 15. Q P R 9 cm xcm Example 75º 75º To find x: 𝑝 𝑠𝑖𝑛𝑃 = 𝑟 𝑠𝑖𝑛𝑅 9 sin75 = 𝑥 𝑠𝑖𝑛75 9 sin75 × 𝑠𝑖𝑛75 = 𝑥 𝒙 = 𝟗 𝒄𝒎 We are given two sides and one angles and so the sine rule can be applied
- 16. A B C b a c Sine Rulefor Angle 𝑠𝑖𝑛𝐴 𝑎 = 𝑠𝑖𝑛𝐵 𝑏 = 𝑠𝑖𝑛𝐶 𝑐
- 17. Q P R 9 cm 4cm Example 75º xº To find x: 𝑠𝑖𝑛𝑃 𝑝 = 𝑠𝑖𝑛𝑅 𝑟 sin75 9 = 𝑠𝑖𝑛𝑥 4 sin75 × 4 9 = 𝑠𝑖𝑛𝑥 0.429300 … = 𝑠𝑖𝑛𝑥 s𝑖𝑛−1 0.429300 = 𝑥 𝒙 = 𝟐𝟓. 𝟒 ̊̊ We are given two angles and one side and so the sine rule can be used
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- 19. COSINE RULE FORSide The cosine rule is a formula which can be used to calculate the missing sides of a triangle or to find a missing angle. To do this we need to know the two arrangements of the formula and what each variable represents.
- 20. The cosine ruleis used to find: a side when two sides and the angle in between them are known an angle when three sides are known COSINE RULE FOR Side
- 21. COSINE RULE FORSide A B C b a c The cosine rule for sides is: a²= b² + c² - (2bc + cosA)
- 22. Example A B C 7 cm xcm 3 cm To find x: a²= b² + c² - (2bc + cosA) a²=7²+3²- (2×3×7+ cos35) a²= 23.5956 cm a= 2 23.5956 cm a= 4.86 cm 35º The cosine rule is used because there are 2 sides AND one angle
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- 24. COSINE RULE FORAngles A B C b a c The cosine rule for angles is: cos(A) = b2 + c2 − a2 2bc
- 25. Example A B C 7 cm 5cm 3 cm To find x: cos(A) = b2 + c2 − a2 2bc cos(A) = 72 + 32 − 52 2 × 7 × 3 cos−1(0.7857) = A A = 38.2̊ xº The cosine rule is used because there are 3 sides AND an unknown angle
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