trigonometric functions section 4.2a homework: p. 368-369 1-39 odd

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Trigonometric FunctionsTrigonometric FunctionsSection 4.2aSection 4.2a

Homework: p. 368-369 1-39 oddHomework: p. 368-369 1-39 odd

Remind me…from geometry:Remind me…from geometry:

What does it mean for two figures to be similar ???

Same shape, but not necessarily same size…

The concept of similar triangles is the basis of righttriangle trigonometry, the topic of this section.

Right Triangle TrigonometryRight Triangle Trigonometry

Standard Position (of an acute angle) – in the xy-plane, thevertex is at the origin, one ray is along the positive x-axis,and the other ray extends into the first quadrant.

Example:

Right Triangle TrigonometryRight Triangle TrigonometryLet be an acute angle in the right . ThenABC

A

C

B

Adj.

Opp

.

Hyp

.

sine sinopp

hyp

cosine cosadj

hyp

tangent tanopp

adj

Right Triangle TrigonometryRight Triangle TrigonometryLet be an acute angle in the right . ThenABC

A

C

B

Adj.

Opp

.

Hyp

.

cosecant cschyp

opp

secant sechyp

adj

cotangent cotadj

opp

Two Famous TrianglesTwo Famous TrianglesThe isosceles right triangle:

1

45

12

Let’s find all 6 trig. functions for a angle.45

1 2sin 45 0.707

22

1 2cos 45 0.707

22

1tan 45 1

1

Two Famous TrianglesTwo Famous TrianglesThe isosceles right triangle:

1

45

12

Let’s find all 6 trig. functions for a angle.45

2csc45 1.414

1

2sec45 1.414

1

1cot 45 1

1

Two Famous TrianglesTwo Famous TrianglesThe 30-60-90 triangle:

1

60

23

Let’s find all 6 trig. functions for a angle.301

sin 302

3cos30 0.866

2

1 3tan30 0.577

33

30

Two Famous TrianglesTwo Famous TrianglesThe 30-60-90 triangle:

1

60

23

Let’s find all 6 trig. functions for a angle.302

csc30 21

2 2 3sec30 1.155

33

3cot 30 1.732

1

30

Two Famous TrianglesTwo Famous TrianglesThe 30-60-90 triangle:

1

60

23

Is there a shortcut for finding the trig.functions for a angle?60

303

sin 602

1

cos602

3tan 60 3

1

Using one trig ratio to find them allUsing one trig ratio to find them all

Let be an acute angle such that . Evaluate theother five trigonometric functions of .

sin 5 6

What does the triangle look like?

6

x

5

Solve for x: 2 26 5 11x

11cos

6

5tan

11

6csc

5

6sec

11 11

cot5

Evaluating Trig Functions with a CalculatorEvaluating Trig Functions with a Calculator

Beware these common errors!!!

2sin

1. Using the calculator in the wrong mode (degrees/radians)

2. Using the inverse trig keys for cot, sec, and csc

Ex: the TAN is not the cotangent function!!!–1

3. Using incorrect shorthand

Ex: to evaluate , you must type 2sin 4. Not closing parentheses

Getting an “exact answer” on a calculatorGetting an “exact answer” on a calculator

Find the exact value of on a calculator.cos30

First, type in Make sure you’re in degree mode!!! cos 30

Next, square your answer You should get 0.75

This suggests that the exact value of is:cos30

3cos30

4

3

2

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