Trig Functions – Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras’ theorem.

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.

Use Trig Functions (RAT) 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

Use Trig Functions (RAT) These are given on page 13 of the Formula and Tables Book, but are presented without the words “opposite”, “adjacent”, or “hypotenuse”.

Use Trig Functions (RAT) e.g. Calculate sin⁡(37), cos⁡(37), and tan⁡(37) using the triangle below:

Use Trig Functions (RAT) 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. Copy the diagram into your answer book and on it mark the angle A such that tan 𝐴 = 5 8 . Find the area of the triangle.

Use Trig Functions (RAT) e.g. Use a calculator to calculate each of the following and fill in the blanks. sin 60 =0.866, so the opposite is 0.866 times as long as the hypotenuse. cos 28 = , so the is times as long as the . tan 76 = , so the is times as long as the . sin 62 = , so the is times as long as the .

Use Trig Functions (RAT) Draw a triangle 𝐴𝐵𝐶 so that 𝐴𝐶 =100, ∠𝐴𝐵𝐶 = 90 𝑜 , and ∠𝐶𝐴𝐵 = 50 𝑜 . Find |𝐴𝐵| and |𝐵𝐶|. Draw a triangle 𝐷𝐸𝐹 so that 𝐷𝐸 =7, ∠𝐷𝐸𝐹 = 90 𝑜 , and ∠𝐹𝐷𝐸 = 25 𝑜 . Find |𝐷𝐹| and |𝐸𝐹|. Draw a triangle 𝐿𝑀𝑁 so that 𝐿𝑀 =62, ∠𝐿𝑀𝑁 = 37 𝑜 , and ∠𝑀𝑁𝐿 = 90 𝑜 . Find |𝐿𝑁| and |𝑀𝑁|.

Use Trig Functions (RAT) 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 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 If cos 𝐴 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 , then 𝐴= cos −1 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 If tan 𝐴 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 , then 𝐴= tan −1 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 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 .

Use Trig Functions (RAT) e.g. Given the triangle shown, find 𝜃: tan (𝜃) = 50 100 ⇒𝜃= tan −1 1 2 ⇒𝜃≈ 26.6 𝑜

Use Trig Functions (RAT) Draw a triangle 𝐴𝐵𝐶 so that 𝐴𝐵 =9, 𝐵𝐶 =4, and ∠𝐶𝐵𝐴 = 90 𝑜 . Find |∠𝐵𝐴𝐶|. Draw a triangle 𝐷𝐸𝐹 so that 𝐷𝐹 =13, 𝐸𝐹 =7.5, and ∠𝐷𝐸𝐹 = 90 𝑜 . Find |∠𝐹𝐷𝐸|. Draw a triangle 𝐿𝑀𝑁 so that 𝐿𝑁 =2.2, 𝐿𝑀 =1.7, and ∠𝐿𝑀𝑁 = 90 𝑜 . Find |∠𝑁𝐿𝑀|.

Use Trig Functions (RAT) 2004 OL P2 Q5a The lengths of the sides of a right-angled triangle are shown in the diagram and 𝐴 is the angle indicated. Write down the value of cos 𝐴 . Hence, find the angle 𝐴, correct to the nearest degree. 2007 OL P2 Q5b In the right-angled triangle Δ𝑎𝑏𝑐, 𝑎𝑏 =5 cm. The area of the triangle is 15 cm2. Find 𝑏𝑐 . Find ∠𝑐𝑎𝑏 to the nearest degree. Find ∠𝑏𝑐𝑎 to the nearest degree.

Use Trig Functions (RAT) 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.

Use Trig Functions (RAT) 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? 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? 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.

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.

Use Pythagoras’ Theorem Find the missing side length in each of the following triangles. 1. 2. 3.

Use Pythagoras’ Theorem 2004 OL P2 Q4a In the triangle 𝑎𝑏𝑐, 𝑎𝑏 =8, 𝑎𝑐 =17 and ∠𝑎𝑏𝑐 = 90 𝑜 . Find |𝑏𝑐|. 2008 OL P2 Q4a In the triangle 𝑎𝑏𝑐, ∠𝑎𝑏𝑐 = 90 𝑜 , 𝑏𝑐 =4.5 and 𝑎𝑐 =7.5. Find |𝑎𝑏|.