Chapter 4

Chapter 4Section 3 Right triangle trigonometry

Objectives• Evaluate trigonometric functions of acute angles • Use fundamental trigonometric identities• Use trigonometric functions

Right triangles• Trigonometry depends on the meaning of similar

figures.• Similar figures are equiangular, and the sides

that make the equal angles are proportional.• PLANE TRIGONOMETRY is based on the fact of

similar figures. •  We saw:• Figures are similar if they are equiangular

and the sides that make the equal angles are proportional.

Right Triangles• Right triangles will be similar if an acute angle of

one is equal to an acute angle of the other.

In the right triangles ABC, DEF, if the acute angle at B is equal to the acute angle at E, then those triangles will be similar.  Therefore the sides that make the equal angles will be proportional.

Right Triangles• Relative to the angle, the three sides of the

triangle are the hypotenuse, the opposite side and the adjacent side

• Using the lengths, you can form six ratios that define the six trigonometric functions:

• Sine, cosine, tangent, cosecant, secant, and cotangent

Right triangle

Example #1• For example, to measure the height h of a

flagpole, we could measure a distance of, say, 100 feet from its base.  From that point P we could then measure the angle required to sight the top .  If that angle, called the angle of elevation, turned out to be 37°, then

Example #2• In right triangle ABC, hypotenuse AB=15

and angle                      A=35º.    Find leg length, BC, to the nearest tenth.

Example #3• In a right triangle, sec θ = 4.  Sketch the triangle,

place the ratio numbers, and evaluate the remaining functions of θ.

Example #4 In a right triangle, cos θ = 

25

.  Sketch the triangle and evaluate sin θ.

Student guided practice

• Do problems 7,8,13,14 in your book page 280

45-45-90 Right triangle

• Is a special case of right triangle where the lengths of the triangle are congruent making the angles congruent.

Example#5• Find the exact values of sin45,cos45 and tan45• Solution:

30-60-90 RIGHT TRIANGLE

• Is another special case of right triangle where the angles of the triangles are 30 and 60.

Example • Use the right triangle to find the exact value of

sin60, cos60,sin30 and cos30.

Sines,cosines and tangent of special

angles

Using a calculator• Use calculator to evaluate• Cos( • Sec(

Trigonometric Identities

Applying trigonometric

identities• Let • A) c• B)s

Example• Let • A) sin• B)tan

Example• Use trigonometric functions to transform one

side of the equation into the other.• A. sec• B.

Student guided practice

• Do problems 33,34,53,59 and 60 in your book 280 and 281

Homework• Do problems 9,10,16,17,55,and 62 in your book

page 280 and 281.

closure• Today we learned about right triangle

trigonometry • Next class we are going to continue with

trigonometry