Lesson 1.1 Lines in the Plane - Miss Cochi's Mathematics

3 . 4 n o s

y s r t e e

L m o n o g

i r T e l g

n a i r T t

h g i R Essential Question: How do you use trigonometry to find unknown side

lengths and angle measures in right triangles? Before we start How would you find the

length of side c? How would you find the length of side x and y? Six Trigonometric Functions

Consider a right triangle, with one acute angle labeled . Relative to the angle, , the three sides of the triangle are the hypotenuse, the opposite

side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). Six Trigonometric Functions

Using the lengths of these three sides, you can form six ratios that define the six trigonometric functions of the acute angle . Sine Cosecant Cosine

Secant Tangent Cotangent Right Triangle Definitions of Trigonometric Functions

The abbreviations opp, adj, and hyp, represent the lengths of the three sides of a right triangle. opp = the length of the side opposite

adj = the length of the side adjacent to hyp = the length of the hypotenuse How do you use trigonometry to find unknown side lengths and angles in right triangles?

If necessary, use Pythagorean theorem to find the 3rd side length. Use the definition of the trigonometric functions to find the ratio of the lengths of the side. Use the definition of the trigonometric functions to find the angle based on your ratio of the

lengths of the side. Use the triangle to find the exact values of the six trigonometric functions of . Use the triangle to find the exact values of the

six trigonometric functions of . Find the exact values of , , and . Use the equilateral triangle shown to find the exact values of , , , and .

Sines, Cosines, and Tangents of Special Angles Sines, Cosines, and Tangents of Special

Angles Note that . This is because 30 and 60 are complementary angles, and, in general, it can be shown from the right triangle definitions that conjunctions of complementary angles are equal. That is, if is an acute angle, then the following relationships are true:

Use a calculator to evaluate . Use a calculator to evaluate . Trigonometric Identities

Reciprocal Identities Trigonometric Identities Quotient Identities Trigonometric Identities

Pythagorean Identities Let be an acute angle such that . Find the values of (a) and (b) using trigonometric identities. Let be an acute angle such that . Find the values

of (a) and (b) using trigonometric identities. Let be an acute angle such that . Find the values of (a) and (b) using trigonometric identities. Verify that .

Verify that . Verify that . Applications

Many applications of trigonometry involve a process called solving right triangles. In this type of application, you are usually given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or you are given two sides and are asked

to find one of the acute angles. Angle of elevation The angle from the horizontal upward to the object.

Angle of depression The angle from the horizontal downward to the object.

A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as . How tall is the tree? A biologist wants to know the width W of a river

in order to properly set instruments for studying the pollutants in the water. From point A, the biologist walks downstream 70 feet and sights to point C. From this sighting, it is determined that . How wide is the river?

You are 200 yards from a river. Rather than walking directly to the river, you walk 400 yards along a straight path to the rivers edge. Find the acute angle between this path and the rivers edge.

A 12-meter flagpole casts a 12-meter shadow. Find , the angle of elevation to the sun. Find the length c of the skateboard ramp. A ramp feet in length rises to a loading platform

that is feet off the ground. Find the angle that the ramp makes with the ground. How do you use trigonometry to find unknown side lengths and angle measures in right triangles?

Ticket Out the Door Find the value of sec

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