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1|Page Chapter 7 Trigonometry Math 1202 2|Page Things to get you started Trigonometry is a branch of Mathematics that allows you to determine the measures of sides and angles in a triangle. In this course we will look at right triangles. Labeling Triangles Angles are always indicated by capital letters Sides are indicated with lower case letters. The angles and their corresponding sides use the same letter. A Side AB same as Side c Side BC same as Side a Side AC same as Side b c b a C B Basics There are three sides to a triangle which can be labelled the hypotenuse, opposite and adjacent sides. The hypotenuse will never change in a right triangle The opposite or adjacent side will depend on the angle you are referring to. The reference angle (usually , theta) will always be one of the acute angles. The opposite side is the side opposite the angle. The adjacent side will be the side that is left (It points to the right angle). Ex) For the following triangle, state : L P M 1) Hypotenuse 2) Side opposite M 3) Side opposite L 4) Side adjacent L 5) Side adjacent M 3|Page Assign Worksheet “Labeling Triangles” Trigonometric Ratios: sin cos tan opposite hypotenuse Read as “Sine of Theta” adjacent hypotenuse Read as “Cosine of Theta” opposite adjacent Read as “Tangent of Theta” There are two ways to remember these ratios: 1) SOHCAHTOA 2) Some Old Hags, Can’t Afford Husbands, Till Old Age Ex) Find the trigonometric ratios for angles A and B in the following triangle. sin A cosA tan A opp hyp 5 13 adj hyp 12 13 opp adj 5 12 4|Page sin B cosB tan B opp hyp 12 13 adj hyp 5 13 opp adj 12 5 Using Calculators to find Trigonometric Ratios You can use any scientific calculator to calculate trigonometric ratios. Be sure your calculator is in degree mode (not radian). Before calculators were readily available, trigonometric tables were used. Always give values of trig ratios to four decimal places. Ex 1) Use a calculator to find the following: A) sin 68º B) tan 47º C) cos 23º Finding sides using Trigonometry You can use trig ratios to find missing sides of a right triangle. Steps 1) Label the sides in terms of “Hyp”, “Opp” and “Adj” relative to the reference angle. 2) For the given angle, determine which trig ratio can be used with the known information. equation based on the trig ratio chosen. 4) Solve the equation using cross multiplication to find the missing side. Note If the unknown value is on top, you multiply to find the missing value. If the unknown value is on bottom, you divide to find the missing value. Ex 2) Determine the value of x in each triangle: A) B) C) 5|Page Finding angles using Trigonometry You can also find missing angles given the trig ratio. You must use sin calculate missing angles. (You may have to press shift, inv…etc first) 1 , cos1 , tan 1 to Ex 1) Use a calculator to find the value of θ: A) sin Ɵ = 0.3265 B) cos Ɵ = 0.5468 C) tan Ɵ = 1.0271 You can use trig ratios to find missing angles of a right triangle. Steps 1) Label the sides in terms of “Hyp”, “Opp” and “Adj” relative to the reference angle. 2) For the given angle, determine which trig ratio can be used with the known information. 3) Write an equation based on the trig ratio chosen. 4) Solve the equation using sin nearest degree) 1 , cos1 or tan 1 to find the missing angle. (Usually to Ex 2) Determine the value of θ in each triangle: A) B) C) 6|Page 4.5 Applications of Trigonometry We have learned how to solve a right triangle given either; 7|Page thelengths of any two sides the length of one side and the measure of an acute angle You can apply these skills to solve word problems involving right triangles. It helps to draw a diagram of the situation so you can determine which trig ratio to use. There are two types of angles mentioned in word problems involving trigonometry. Angle of Depression Angle of Elevation Ex 1) An 8 m ladder is placed against the wall so that it makes an angle of elevation of 32̊ with the ground. How far is the base of the ladder from the wall? x 8 x 8 cos32o x 6 .8 cos32o 8m 32° x The base of the ladder is 6.8 m from the wall. 8|Page Ex 2) A plane is flying at an altitude of 450 m. It lands at an angle of depression of 28̊. How far does the plane fly before it lands? 28° 450 m 450 x 0 .4695 450 1 x 0 .4695x 450 450 x 0 .4695 x 958 sin 28 28° The plane flies 958 m. Using Trigonometry to find missing sides in a right triangle