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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
