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Right Triangle Trigonometry. Objectives. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle measures in right triangles. History . Right triangle trigonometry is the study of the relationship between the sides and angles of

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Presentation Transcript
objectives
Objectives
  • Find trigonometric ratios using right triangles.
  • Use trigonometric ratios to find angle measures in right triangles.
history
History

Right triangle trigonometry is the study of the

relationship between the sides and angles of

right triangles. These relationships can be used

to make indirect measurements like those using

similar triangles.

trigonometric ratios

Trigonometric Ratios

Only Apply to Right Triangles

the 3 trigonometric ratios
The 3 Trigonometric Ratios
  • The 3 ratios are Sine, Cosine and Tangent
slide7

θ

The six trigonometric functions of a right triangle, with an acute angle ,are defined by ratios of two sides of the triangle. The sides of the right triangle are:

hyp

 the side opposite the acute angle ,

opp

 the side adjacent to the acute angle ,

adj

 and the hypotenuse of the right triangle.

The trigonometric functions are

sine, cosine, tangent, cotangent, secant, and cosecant.

evaluating trigonometric functions
EVALUATING TRIGONOMETRIC FUNCTIONS

What is the value of h?

Find all six trig functions of angle A

Remember SOH CAH TOA and the reciprocal identities

evaluating trigonometric functions1
EVALUATING TRIGONOMETRIC FUNCTIONS

What is the value of h?

Find all six trig functions of angle A

Remember SOH CAH TOA and the reciprocal identities

evaluating trigonometric functions2
EVALUATING TRIGONOMETRIC FUNCTIONS

What is the value of b?

Find all six trig functions of angle A

Remember SOH CAH TOA and the reciprocal identities

slide11

1

sin 45 = = =

cos 45 = = =

45

1

tan 45 = = = 1

cot 45 = = = 1

adj

opp

hyp

adj

hyp

hyp

adj

sec 45 = = =

csc 45 = = =

opp

adj

opp

Calculate the trigonometric functions for a 45 angle.

slide12

2

2

60○

60○

2

Geometry of the 30-60-90 triangle

Consider an equilateral triangle with each side of length 2.

30○

30○

The three sides are equal, so the angles are equal; each is 60.

The perpendicular bisector of the base bisects the opposite angle.

1

1

Use the Pythagorean Theorem to find the length of the altitude, .

slide13

2

1

30

Calculate the trigonometric functions for a 30 angle.

slide14

2

1

60

Calculate the trigonometric functions for a 60 angle.

trig functions complements
TRIG FUNCTIONS & COMPLEMENTS

Two positive angles are complements if the sum of their measures is .

Example: are complement because .

The sum of the measures of the angles in a triangle is . In a right triangle, we have a angle. That means that the sum of the other two angles is . Those two angles are acute and complement.

If the degree measure of one acute angle is , then the degree measure of the other angle is .

trig functions complements1
TRIG FUNCTIONS & COMPLEMENTS

Compare and .

Therefore, . If two angles are complements, the sine of one equals the cosine of the other.

using cofunction identities
Using cofunction identities

Find a cofunction with the same value as the given expression:

Find a cofunction with the same value as the given expression:

slide24

A surveyor is standing 50 feetfrom the base of a large tree. The surveyor measures the angle of elevationto the top of the tree as 71.5°. How tall is the tree?

tan 71.5°

?

tan 71.5°

71.5°

y = 50 (tan 71.5°)

50

y = 50 (2.98868)

Look at the given info. What trig function can we use?

slide25

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

200

60°

x

x

X = 400 yards

Look at the given information. Which trig function should we use?

slide26

A guy wire from a point 2 m from the top of an electric post makes an angle of 700 with the ground. If the guy wire is anchored 5 m from the base of the post, how high is the pole?

2 m

h = (13.74 + 2) meters

Guy wire

x

700

h = 15.74 meters

5 m

Which trig function should we use?