1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem.
Opposite Adjacent 2 Use Trig Functions (Right-Angled Triangles) Recall in a right-angled triangle, we name three sides based on the angle of interest: Hypotenuse is always the longest side. Opposite is literally opposite the angle of interest. Adjacent is literally adjacent to the angle of interest. A A Adjacent Opposite
3 For a particular angle, the ratio between the side lengths is the same for every triangle, regardless of size. The ratio of the opposite to the hypotenuse is called sine: sin A = opposite hypotenuse The ratio of the adjacent to the hypotenuse is called cosine: cos A = adjacent hypotenuse The ratio of the opposite to the adjacent is called tangent: tan A = opposite adjacent
4 These are given on page 13 of the Formula and Tables Book, but are presented without the words opposite, adjacent, or hypotenuse.
5 e.g. Calculate sin(37), cos(37), and tan(37) using the triangle below:
6 2003 OL P2 Q5a The lengths of the sides of a right-angled triangle are shown in the diagram and B is the angle indicated. Find the value of sin B cos B, as a fraction. 2006 OL P2 Q5a The lengths of two sides of a right-angled triangle are shown in the diagram. i. Copy the diagram into your answer book and on it mark the angle A such that tan A = 5 8. ii. Find the area of the triangle.
7 e.g. Use a calculator to calculate each of the following and fill in the blanks. 1. sin 60 = 0.866, so the opposite is 0.866 times as long as the hypotenuse. 2. cos 28 =, so the is times as long as the. 3. tan 76 =, so the is times as long as the. 4. sin 62 =, so the is times as long as the.
8 1. Draw a triangle ABC so that AC = 100, ABC = 90 o, and CAB = 50 o. Find AB and BC. 2. Draw a triangle DEF so that DE = 7, DEF = 90 o, and FDE = 25 o. Find DF and EF. 3. Draw a triangle LMN so that LM = 62, LMN = 37 o, and MNL = 90 o. Find LN and MN.
9 While the trig functions apply to angles and return the ratio of side lengths, there are inverse trig functions that apply to ratios and return angles. If sin(a) = opposite hypotenuse, then A = sin 1 opposite If cos A = adjacent adjacent, then A = cos 1 hypotenuse If tan A = opposite, then A = tan 1 opposite adjacent hypotenuse hypotenuse adjacent On a calculator, these are typed using e.g. SHIFT+sin sin 1 B is pronounced inverse sine of B or sine minus one of B, likewise for cos 1 B and tan 1 B.
10 e.g. Given the triangle shown, find θ: tan(θ) = 50 100 θ = tan 1 1 2 θ 26.6 o
11 1. Draw a triangle ABC so that AB = 9, BC = 4, and CBA = 90 o. Find BAC. 2. Draw a triangle DEF so that DF = 13, EF = 7.5, and DEF = 90 o. Find FDE. 3. Draw a triangle LMN so that LN = 2.2, LM = 1.7, and LMN = 90 o. Find NLM.
12 2004 OL P2 Q5a 2007 OL P2 Q5b The lengths of the sides of a right-angled triangle are shown in the diagram and A is the angle indicated. i. Write down the value of cos A. ii. Hence, find the angle A, correct to the nearest degree. In the right-angled triangle Δabc, ab = 5 cm. The area of the triangle is 15 cm 2. i. Find bc. ii. iii. Find cab to the nearest degree. Find bca to the nearest degree.
13 In real world problems, two additional terms are important. When looking up, the angle from the horizontal is called the angle of elevation. When looking down, the angle from the horizontal is called the angle of depression. Additionally, a clinometer is a device used to measure these angles.
14 1. From the top of a light house 60 meters high with its base at the sea level, the angle of depression of a boat is 15 degrees. What is the distance of boat from the foot of the light house? 2. The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 m from its base is 45 degrees. If the angle of elevation of the top of the complete pillar at the same point is to be 60 degrees, then the height of the incomplete pillar is to be increased by how much? 3. A 10-metre-long ladder rests against a vertical wall so that the distance between the foot of the ladder and the wall is 2 metres. Find the angle the ladder makes with the wall and height above the ground at which the upper end of the ladder touches the wall.
15 Use Pythagoras Theorem Pythagoras Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
16 Use Pythagoras Theorem Find the missing side length in each of the following triangles. 1. 2. 3.
17 2004 OL P2 Q4a 2008 OL P2 Q4a Use Pythagoras Theorem In the triangle abc, ab = 8, ac = 17 and abc = 90 o. Find bc. In the triangle abc, abc = 90 o, bc = 4.5 and ac = 7.5. Find ab.