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**Trigonometry Sine Rule**

A B C a c b P Q R p q r Mr Porter

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**Definition: The Sine Rule**

In any triangle ABC ‘The ratio of each side to the sine of the opposite angle is CONSTANT. A B C a c b P Q R p q r For triangle ABC For triangle PQR

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**Example 2: Find the size of α in ∆PQR in degrees and minutes.**

Example 1: Use the sine rule to find the value of x correct to 2 decimal places. Example 2: Find the size of α in ∆PQR in degrees and minutes. When to apply the sine rule: Is there 2 sides and 2 angles (or more)? YES, then use the Sine Rule. When to apply the sine rule: Is there 2 sides and 2 angles (or more)? YES, then use the Sine Rule. C A B 8.3 cm x 81° 43° Q P R 22.4 cm 14 cm 118°27’ α Label the triangle. p q r Label the triangle. a b c Write down the sine rule for this triangle Write down the sine rule for this triangle We do not need the ‘Q’ ratio. Substitute P, p, R and r. Rearrange to make sinα the subject. We do not need the ‘C’ ratio. Substitute A, a, B and b. Rearrange to make x the subject. Evaluate RHS Use calculator. To FIND angle, use sin-1 (..) Convert to deg. & min.

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**Ambiguous Case – Angles (Only) **

But, could the triangle be drawn a different way? The answer is YES! Example 1 : Use the sine rule to find the size of angle θ. C B A 41° 14.5 cm 9.8 cm θ C B A 41° 14.5 cm 9.8 cm θ A 9.8 cm θ Label the triangle. a b c When to apply the sine rule: Is there 2 sides and 2 angles (or more)? YES, then use the Sine Rule. Write down the sine rule for this triangle to find an angle By supplementary angles: We do not need the ‘C’ ratio. Substitute A, a, B and b. Rearrange to make sin θ the subject. Which is correct, test the angle sum to 180°, to find the third angle, α. Case 1: Evaluate RHS Case 2: To FIND angle, use sin-1 (..) Hence, both answers are correct!

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**Ambiguous Case – Angles (Only) **

But, could the triangle be drawn a different way? The answer is NO! Example 1 : Use the sine rule to find the size of angle θ. Lets check the supplementary angle METHOD. Q P R 122° θ 17 cm 8.5 cm When to apply the sine rule: Is there 2 sides and 2 angles (or more)? YES, then use the Sine Rule. Label the triangle. p q r Which is correct, test the angle sum to 180°, to find the third angle, α. Write down the sine rule for this triangle Case 1: We do not need the ‘P’ ratio. Substitute Q, R, q and r. Rearrange to make sin θ the subject. Case 2: Evaluate RHS Hence, the ONLY answer is correct! To FIND angle, use sin-1 (..)

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d H B L 4000 m 60° 50° 70° Example 3: Points L and H are two lighthouses 4 km apart on a dangerous rocky shore. The shoreline (LH) runs east–west. From a ship (B) at sea, the bearing of H is 320° and the bearing of L is 030°. a) Find the distance from the ship (B) to the lighthouse (L), to the nearest metre. b) What is the bearing of the ship (B) from the lighthouse at H? Label the triangle. l b h When to apply the sine rule: Is there 2 sides and 2 angles (or more)? YES, then use the Sine Rule. N 0° H L 4 km B (ship) Write down the (side) sine rule for this triangle (a) 320° 50° 30° 60° 30° We do not need the ‘L’ ratio. Substitute B, H, b and d, (h). Rearrange to make d the subject. 70° 40° Use calculator. 50° (b) From the original diagram : Use basic alternate angles in parallel line, Bearing and angle sum of a triangle to find all angles with the ∆BHL. Bearing of Ship from Lighthouse H: H = 90°+50° H = 140° Re-draw diagram for clarity

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Example 4: To measure the height of a hill a surveyor took two angle of elevation measurements from points X and Y, 200 m apart in a straight line. The angle of elevation of the top of the hill from X was 5° and from Y was 8°. What is the height of the hill, correct to the nearest metre? Use calculator Do NOT ROUND OFF! h X Y B T 200 m 5° 8° To find h, use the right angle triangle ratio’s i.e. sin θ. To find ‘h’, we need either length BY or TY! 3° 82° When to apply the sine rule: Is there 2 sides and 2 angles (or more)? YES, then use the Sine Rule to find TY= x. x 172° Use basic angle sum of a triangle, exterior angle of a triangle and supplementary angles to find all angles with the ∆AYT And ∆YTB. Hence, the hill is 46 m high (nearest metre). Write down the (side) sine rule for this triangle We do not need the ‘Y’ ratio. Substitute X, T, t and x. Rearrange to make x the subject.

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Example 5: A wooden stake, S, is 13 m from a point, A, on a straight fence. SA makes an angle of 20° with the fence. If a hores is tethered to S by a 10 m rope, where, on the fence, is the nearest point to A at which it can graze? Evaluate RHS To FIND angle, use sin-1 (..) Fence • Stake A B C S 20° 10 m 13 m From the diagram, it is obvious that angle B is Obtuse. B = 180 – 26° 24’ B = 153° 36’ Then angle ASB = 182 – (153°36’ + 20) = 6° 24’ Now, apply the sine rule to find the length of AB. Write down the (side) sine rule for this triangle 1) The closest point to A along the fence, is point B. Hence, we need to find distance AB. 2) Look at ∆ABS, to use the sine rule, need to find angle ABS or angle ASB. Write down the (angle) sine rule for this triangle We do not need the ‘S’ ratio. Substitute A, B, a and b. Rearrange to make sin B the subject. AB = m

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**[There are several different solution!]**

Example 6: Q, A and B (in that order) are in a straight line. The bearings of A and B from Q is 020°T. From a point P, 4 km from Q in a direction NW, the bearing of A and B are 112°T and 064°T respectively. Calculate the distance from A to B. In ∆POA, write down the (side) sine rule for this triangle d 112° N 0° P A 4 km Q B 45° 20° 64° Not to scale! x y We do not need the ‘P’ ratio. Substitute Q, A, q and y. Rearrange to make y the subject. Use calculator Do NOT ROUND OFF! 23° 45° 48° 92° 44° 88° In ∆PAB, write down the (side) sine rule. For this triangle. We do not need the ‘A’ ratio. Substitute P, B, y and d. Rearrange to make d the subject. Use basic alternate angles in parallel line, Bearing and angle sum of a triangle to find all angles in the diagram. Use calculator and y = Do NOT ROUND OFF! To find ‘d’, we need to work backward using the sine rule, meaning that we must find either x or y first. [There are several different solution!]

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