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Warm – up Find the missing measures. Write all answers in radical form. y 60° 30 10 45 y x 60 45 z 30° z x6 y3 3 z 5 3 z3 2 y 5 Introduction to Trigonometry This section presents the 3 basic trigonometric ratios sine, cosine, and tangent. The concept of similar triangles and the Pythagorean Theorem can be used to develop the trigonometry of right triangles. Engineers and scientists have found it convenient to formalize the relationships by naming the ratios of the sides. You will memorize these 3 basic ratios. The Trigonometric Functions SINE COSINE TANGENT SINE Pronounced like “sign” COSINE Pronounced like “co-sign” TANGENT Pronounced “tan-gent” B Opp to A Sin(A) With Respect to angle A, Hyp label the three sides Adj to A Cos(A) Hyp A C Opp to A Tan(A) Adj to A We need a way to remember all of these ratios… SOHCAHTOA Sin Opp Hyp Cos Adj Hyp Tan Opp Adj Finding sin, cos, and tan. (Just writing a ratio or decimal.) Find the sine, the cosine, and the tangent of M. Give a fraction and decimal answer (round to 4 places). N opp 9 sin M hyp 10.8 .8333 9 10.8 adj 6 cos M hyp 10.8 .5556 P 6 M opp 9 tan M 1.5 adj 6 A Find the sine, 24.5 cosine, and the 8.2 tangent of angle A C 23.1 B opp 23.1 sin A .9429 Give a fraction hyp 24.5 and decimal answer. adj 8 .2 cos A .3347 hyp 24.5 Round to 4 decimal places opp 23.1 tan A 2.8171 adj 8.2 Finding a side. (Figuring out which ratio to use and getting to use a trig button.) Ex: 1 Find x. Round to the nearest tenth. Figure out which ratio to use. 55 What we’re looking for… What we know… 20 m adj opp tan adj x tan55 x opp 20 20 tan55 x 20 tan 55 ) x 28.6 m We can find the tangent of 55 using a calculator Ex: 2 Find the missing side. Round to the nearest tenth. 283 m sin 24 x x 24 283 283 sin 24 x x 115.1 m Ex: 3 Find the missing side. Round to the nearest tenth. cos40 x 40 20 m x 20 20 cos40 x x 15.3 m Ex: 4 Find the missing side. Round to the nearest tenth. tan72 80 x 80 m x tan72 80 Note: When the variable is x 80 72 in the denominator, tan(72 ) x you end up dividing 80 tan 72 ) = x 26 m Sometimes the right triangle is hiding ABC is an isosceles triangle as marked. Find sin C. Answer as a fraction. A 13 13 12 opp 12 sin C B C hyp 13 10 5 Ex. 5 A person is 200 yards from a river. Rather than walk directly to the river, the person walks along a straight path to the river’s edge at a 60° angle. How far must the person walk to reach the river’s edge? cos 60° 200 x (cos 60°) = 200 60° x x X = 400 yards Ex: 6 A surveyor is standing 50 metres from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree? Opp tan 71.5° Adj y ? tan 71.5° 50 71.5° 50 (tan 71.5°) = y 50 m y 149.4 m For some applications of trig, we need to know these meanings: angle of elevation and angle of depression. Angle of Elevation If an observer looks UPWARD toward an object, the angle the line of sight makes with the horizontal. Angle of elevation Angle of Depression If an observer looks DOWNWARD toward an object, the angle the line of sight makes with the horizontal. Angle of depression Finding an angle. (Figuring out which ratio to use and getting to use the 2nd button and one of the trig buttons. These are the inverse functions.) Ex. 1: Find . Round to four decimal places. 17.2 tan 9 17.2 2nd tan 17.2 9 ) 9 62.3789 Make sure you are in degree mode (not radians). Ex. 2: Find . Round to three decimal places. 7 7 cos 23 23 2nd cos 7 23 ) 72.281 Make sure you are in degree mode (not radians). Ex. 3: Find . Round to three decimal places. sin 200 200 400 2nd sin 200 400 ) 30 Make sure you are in degree mode (not radians). When we are trying to find a side we use sin, cos, or tan. When we need to find an angle we use sin-1, cos-1, or tan-1.