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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.

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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.

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The Trigonometric Functions SINE COSINE TANGENT

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SINE Pronounced like “sign” Pronounced like “co-sign” COSINE Pronounced “tan-gent” TANGENT

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A B C With Respect to angle A, label the three sides

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We need a way to remember all of these ratios…

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SOHCAHTOA Sin Opp Hyp Cos Adj Hyp Tan Opp Adj

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Finding sin, cos, and tan. (Just writing a ratio or decimal.)

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Find the sine, the cosine, and the tangent of M. Give a fraction and decimal answer (round to 4 places). 9 6 10.8 M P N

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Find the sine, cosine, and the tangent of angle A A 24.5 23.1 8.2 Give a fraction and decimal answer. Round to 4 decimal places B C

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Finding a side. (Figuring out which ratio to use and getting to use a trig button.)

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Ex: 1 Find x. R ound to the nearest tenth. 20 m x tan 2055 ) Figure out which ratio to use. What we’re looking for… opp What we know… adj We can find the tangent of 55 using a calculator

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Ex: 2 Find the missing side. Round to the nearest tenth. 283 m x

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Ex: 3 Find the missing side. Round to the nearest tenth. 20 m x

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Ex: 4 Find the missing side. Round to the nearest tenth. 80 m x tan 8072 = ) Note: When the variable is in the denominator, you end up dividing

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hidingSometimes the right triangle is ABC is an isosceles triangle as marked. Find sin C. A 13 10 Answer as a fraction. B C 5 12

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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? 200 x Ex. 5 60° cos 60° x (cos 60°) = 200 x X = 400 yards

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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? 50 m ? tan 71.5° 50 (tan 71.5°) = y Ex: 6 y 149.4 m 71.5°

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For some applications of trig, we need to know these meanings: angle of elevation and angle of depression.

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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 Elevation

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Angle of Depression If an observer looks DOWNWARD toward an object, the angle the line of sight makes with the horizontal. Angle of depression

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Finding an angle. (Figuring out which ratio to use and getting to use the 2 nd button and one of the trig buttons. These are the inverse functions.)

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Ex. 1: Find . Round to four decimal places. 9 17.2 Make sure you are in degree mode (not radians). 2 nd tan 17.29)

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Ex. 2: Find . Round to three decimal places. 23 7 Make sure you are in degree mode (not radians). 2 nd cos 723)

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Ex. 3: Find . Round to three decimal places. 400 200 Make sure you are in degree mode (not radians). 2 nd sin 200400)

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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.

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