Introduction to Trigonometry This section presents the 3

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

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 9 10. 8 P 6 M

B 8. 2 C Find the sine, cosine, and the tangent of angle A 24. 5 23. 1 Give a fraction and decimal answer. Round to 4 decimal places A

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. What we’re looking for… What we know… 20 m adj x opp 20 We can find the tangent of 55 using a calculator tan 55 )

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

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

Ex: 4 side. Find the missing Round to the nearest tenth. 80 m Note: When the variable is in the denominator, you end up dividing x 80 tan 72 ) =

Sometimes the right triangle is hiding ABC is an isosceles triangle as marked. Find sin C. Answer as a fraction. A 13 13 12 B 10 C 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° 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? tan 71. 5° ? 71. 5° 50 m 50 (tan 71. 5°) = y 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 2 nd button and one of the trig buttons. These are the inverse functions. )

Ex. 1: Find . Round to four decimal places. 17. 2 9 nd 2 tan 17. 2 9 ) Make sure you are in degree mode (not radians).

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

Ex. 3: Find . Round to three decimal places. 200 0 0 4 nd 2 sin 200 400 ) 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.