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RIGHT-ANGLED TRIGONOMETRY Higher Tier RIGHT ANGLED TRIGONOMETRY Trigonometry in a right-angled triangle can be used to calculate unknown sides and angles. Labelling the sides of a Triangle The hypotenuse (hyp) is always opposite the right angle. The opposite (opp) is always opposite the angle given. The adjacent (adj) is always next to the angle given. The Trigonometric Ratios The three trigonometric ratios are These ratios can be remembered by using the word: These formula triangles are also useful: sohcahtoa ©RSH 13 April 2010 Page 1 of 7 RIGHT-ANGLED TRIGONOMETRY Higher Tier Using your calculator Make sure that your calculator is set to the degree mode, deg or something similar should be displayed. Examples sin 30° = 05 sin 60° = 08660 cos 45° =07071 tan 70° = 27475 You also need to reverse the process by using the SHIFT or INV or 2nd F button. Examples sin1 0.5 30 cos1 0.7071 45 tan1 2.25 66 sohcahtoa ©RSH 13 April 2010 Page 2 of 7 RIGHT-ANGLED TRIGONOMETRY Higher Tier Finding the size of an angle Step 1 Label the sides Step 2 Decide on the ratio to use Step 3 Put in the numbers Step 4 Change the fraction to a decimal Step 5 Find the angle EXAMPLE Find the size of angle SOLUTION Label the sides. The opposite and the hypotenuse are given, so use sine. sin opp hyp 8 12 0.6667 sin1 0.6667 41.8 Important Don’t round your answers until the very end of the calculation. sohcahtoa ©RSH 13 April 2010 Page 3 of 7 RIGHT-ANGLED TRIGONOMETRY Higher Tier Finding the length of a side EXAMPLE 1 Calculate the length of side AB. SOLUTION AB is opposite the given angle. The hypotenuse is also given. Sine is used in this case. opp hyp AB 12 AB 12 sin 40 7.71 (2 d.p.) sin 40 EXAMPLE 2 Calculate the length of AC. SOLUTION The angle and the adjacent are given. AC is the hypotenuse. Cosine is used in this case. AC adj cos 4.5 cos 25 4.97 sohcahtoa ©RSH 13 April 2010 Page 4 of 7 RIGHT-ANGLED TRIGONOMETRY Higher Tier APPLICATIONS There are many applications of trigonometry – navigation, surveying, building design etc. Angles of Elevation and Depression The angle of elevation is measured from the horizontal upwards. The angle of depression is measured from the horizontal downwards. sohcahtoa ©RSH 13 April 2010 Page 5 of 7 RIGHT-ANGLED TRIGONOMETRY Higher Tier EXAMPLE 1 From a point 55 m away from the base of a building, a man measures the angle of elevation to the top of the building as 73°. How tall is the building ? SOLUTION Draw a sketch. The angle is given, so is the adjacent. The opposite is unknown. h 55 tan73 180 m (round answers sensibly) EXAMPLE 2 From the top of a 28 m cliff, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff ? SOLUTION 28 d 28 d 48.5m tan30 tan30 sohcahtoa ©RSH 13 April 2010 Page 6 of 7 RIGHT-ANGLED TRIGONOMETRY Higher Tier Navigation EXAMPLE A ship travels due East from A to B, a distance of 74 km. It then travels due South to C. C is 47 km from B. a. What is the bearing of C from A? b. What is the distance from A to C SOLUTION a. The required bearing is . = 90° + angle BAC. 47 74 angle BAC = 32° (to the nearest degree) tanBAC = 90° + 32° = 122° b. Using Pythagoras’ theorem AC² = AB² + BC² AC² = 74² + 47² = 7685 AC = 88 km (to the nearest km) sohcahtoa ©RSH 13 April 2010 Page 7 of 7