Unit No: F3HW 11. Unit Title: Maths Craft 2. 4 Trigonometry Sine and Cosine Rules. Engineering and Construction

Transcription

1 Unit No: F3HW 11 Unit Title: Maths Craft 4 Trigonometry Sine and Cosine Rules

2 SINE AND COSINE RULES TRIGONOMETRIC RATIOS Remember: The word SOH CAH TOA is a helpful reminder. In any right-angled triangle, the trigonometric ratios are defined as follows: sin x opposite hypotenuse adjacent cos x hypotenuse opposite tan x adjacent O A O i.e. sin x --- cos x --- tan x --- H H A The ratios are used in Mathematics to calculate the lengths of sides or angles in and RIGHT-ANGLED triangle as seen in the examples overleaf.

3 Example 1 Find x In relation to the angle, we are given the opposite and hypotenuse. Here we must use the sine ratio. O 7 sin x H 10 so x sin -1 (0.7) ( d.p.) Example Find d In relation to the angle this time, the sides give opposite and adjacent. We must use the tangent ratio. O tan x --- A tan 3 d 4 > tan3 x d 4 d 4 tan3 > d 6.4 cm All practical examples where there is a right-angled triangle can be completed in this way. However problems arise if there is no right angle. By dividing a diagram into right-angled triangles, we can make use of the ratios but it is time-consuming as in Example 3. 3

4 Example 3 Calculate the length of BC. Using the method above, we must first form right-angled triangles. Now find the height, h. From ABD h sin 40 8 h 8 x sin cm Now to calculate BC, from CBD: sin 70 BC h > sin BC BC x sin BC 5.14 sin cm A much easier and quicker method is to use the Sine Rule. The Sine Rule The Sine Rule states that in any ABC, a b c sin A sin B sin C 4

5 Note: If the angles of the triangle are named with the capital letters A, B and C, then the opposite sides are named with the corresponding lower case letters a, b and c respectively. Sometimes to make the calculation easier, the sine rule may be used in the inverted form: The proof of the Sine Rule is as follows: sin A sin B sin C a b c Angle at A is acute In triangle ACD h sin A b so h asinb In triangle BCD sin B a h so h asinb Angle at A is obtuse In triangle ACD h sin (180 - A) b but sin (180 - A) sina In triangle BCD sin B a h so h asin B In both cases h bsin A h asin B so bsina asin B (This avoids having to calculate the altitude). becomes so or bsina asinb bsina asinb sina sinb sina sinb b sinb a sina a sina a sinb (by dividing each side by sina sinb) 5

6 By starting with a different altitude, you could prove that a sina c sinc In any triangle ABC, a sina b sinb c sinc This is called the sine rule. Example 1 Calculate the length or RQ. Step 1. Write down the sine rule for the triangle Tick off the values which you are given. So use the ratios involving p and q only. Step. p 8 sin 5 sin 75 To simplify this equation, where two fractions are equal is to cross-multiply. This involves multiplying the numerator on each side by the opposite denominator and equating the results. Here p x sin 75 8 x sin 5 8 x sin 75 8 x 0.43 p 3.50 cm ( d.p.) sin i.e. The length of RQ is 3.50 cm 6

7 Example Two observers see an object in the sky. It is 400m from observer A and 30m from observer B. The angle of elevation from B is 64, find the angle of elevation from A. Solution Draw a diagram. Write down the sine rule The angle of elevation from observer A is 46.0 Now try the following exercise. EXERCISE 1 1. Calculate the length of AB.. Find the length of QR. 3. Find the length of AB. (Remember that the sum of the angles of a triangle is 180 ) 4. Calculate the size of the angle at P. 7

8 5. Calculate the size of the angle at C. There are certain triangles where it is not possible to use the sine rule immediately. Example 3 When applying the sine rule, we get i.e. p sin p 14 sin P q sin Q q sin 15 r sin R 18 sin R As you can see, there is not enough information to use the rule. (The numerator and denominator must BOTH be given). To solve this problem, we have to use another rule the Cosine Rule 8

9 The Cosine Rule The Cosine Rule states that any ABC, a b + c b c cos A The proof is as follows: Angle at A is acute Using Pythagoras Theorem in triangle BCD a h + (c - x) a h + c + x cx a (h + x ) + c cx b h + x a b + c cx x cosa x bcosa b so a b + c bcosa Angle at A is obtuse Using Pythagoras Theorem in triangle BCD. a h + (c + x) a h + c + x + cx a (h + x ) + c + cx b h + x a b + c + cx x cos(180 A) -cosa x -bcosa b so a b + c bcosa In any triangle ABC, a b + c bcosa This is called the cosine rule. 9

10 Example 3 The length of PR is 8.44 (to d.p.) Now do the following exercise. EXERCISE 1. Using the cosine rule, complete b... Calculate the length of LN. 3. Calculate the length of AB, 10

11 Using the Cosine Rule given 3 sides Sometimes the Cosine Rule is used to find an angle in a triangle when all three sides are given. The Cosine Rule equation is re-arranged to make the angle the subject. cos A b + c a bc Example 4 Calculate the largest angle in the triangle ABC AC is the longest side and so B will be the largest angle. Stating the Cosine rule: cos B a + c b a c cos B x 5 x (Note: The value is negative because the angle is OBTUSE and the cosine value of an obtuse angle is negative. You will learn more about this later). EXERCISE 3 B (d.p.) The largest angle in the triangle is Using the cosine rule, complete cosr.. Calculate the size of angle N. 11

12 3. Calculate the size of angle R. Remember that cosines of angles between 90 and 180 are negative. 4. Calculate the largest angle in triangle ABC. Sine or Cosine Rule In the next exercise you select the correct rule to find the solutions. EXERCISE 4c Select the correct rule and complete the following questions: 1. Calculate the length of PR.. Calculate the size of angle C. 3. Calculate the size of angle L. 4. Calculate the length of QR. EXERCISE 4 Answers Exercise 1 1) cm ) 9.58 cm 3) 8.57 cm 4) ) Exercise 1) b a + c ac cos B ) 4.17 cm 3) 1.80 cm p + q r Exercise 3 1) cos R ) ) 9.87 cm 4) 8.8 pq Exercise 4 1) 4.5 cm ) ) )10.5c m 1

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