4.3 - r ight t riangle t rigonometry. i n this section, you will learn to : evaluate trigonometric...
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4.3 - RIGHT TRIANGLE
TRIGONOMETRY
IN THIS SECTION, YOU WILL LEARN TO:
Evaluate trigonometric functions of
acute angles Use fundamental trigonometric identities Use a calculator to evaluate trigonometric functions
SIX TRIGONOMETRIC FUNCTIONS OF A RIGHT TRIANGLE:
Use SOH CAH TOA to remember all six of the trigonometric functions:
sin csc
cos sec
tan cot
opp hypA A
hyp opp
adj hypA A
hyp adj
opp adjA A
adj opp
EXAMPLE:
12
Given a right triangle below, find the values
of the six trigonometric functions of :
12
6
ANSWERS:
Use the Pythagorean Theorem to find the hypotenuse:
6
12
6 5sin
56 5
2 2 26 12 6 5c c
6 5
5sec
2
12 2 5cos
56 5
6 1tan
12 2
csc 5
cot 2
SPECIAL RIGHT TRIANGLES:
45 45 90 and 30 60 90
SIX TRIGONOMETRIC VALUES OF THE SPECIAL RIGHT TRIANGLES:
* Note that certain sine and cosine angles are equal.
1 2 3sin30 sin 45 sin 60
2 2 2
3 2 1cos30 cos45 cos60
2 2 2
3tan30 tan 45 1 tan 60 3
3
COFUNCTIONS OF COMPLEMENTARY ANGLES ARE CONGRUENT:
• Sine-Cosine, Tangent-Cotangent, and Secant-Cosecant are cofunctions.
sin 90 cos cos 90 sin
tan 90 co cot 90 tant
sec 90 csc csc 90 sec
EXAMPLE OF COMPLEMENTARY ANGLES: • Use the given function values and the cofunction relationships to find the indicated trigonometric functions.
3 1Given sin 60 and cos60 ,
2 2
find tan 60 , cos30 and sin30 .
ANSWERS:
3cos30 sin 90 30 sin 60
2
tan 60 cot 90 60 cot30 3
1sin30 cos 90 30 cos60
2
FUNDAMENTAL TRIGONOMETRIC IDENTITIES:
A) Reciprocal Identities:
1 1sin csc
csc sin1 1
cos secsec cos
1 1tan cot
cot tan
FUNDAMENTAL TRIGONOMETRIC IDENTITIES:
B) Quotient Identities:
sintan
cos
coscot
sin
FUNDAMENTAL TRIGONOMETRIC IDENTITIES:
C) Pythagorean Identities:
2 2
2 2
2 2
sin cos 1
tan 1 sec
cot 1 csc
PROOF OF THE SINE PYTHAGOREAN IDENTITY:
Prove: 2 2sin cos 1
x
y
r
2 2
1y x
r r
2 2
2 21
y x
r r
2 2 2x y r
EXAMPLES:
1) Use trigonometric identities to find the
values of sin and tan given cos 0.24.
ANSWERS:
1) Given : cos 0.24 2 2sin cos 1
22sin 0.24 1 2sin 0.9424
sin 0.97
sin 0.97tan 4.04
cos 0.24
EXAMPLES:
2) Use trigonometric identities to transform the left side of the equation into the right side.
sin coscsc sec
cos sin
ANSWERS:
2 2sin coscsc sec
sin cos sin cos
sin cos2) Prove the identity : csc sec
cos sin
Find a common denominator :
2 2sin coscsc sec
sin cos
1csc sec
sin cos
csc sec csc sec
CALCULATOR EXAMPLES:
1) Use a calculator to evaluate sec 45 2' 3''.
ANSWERS::1) Since most calculators do not have the
second notation, first change this sec 45 2' 3''
DMS into a decimal form before you find the
value.
1
1Use your calculator as: sec 45.0342
cos 45.0342
or cos 45.0342 1.415x
2 3sec 45 2' 3'' sec 45 sec 45.0342
60 3600
CALCULATOR EXAMPLES:
2) Using a calculator, find the value in
degrees and radians of tan 0.1234.
ANSWERS:
2) Using a calculator, find tan 0.1234.
1
1
Degree mode : tan 0.1234 7.0348
Radian mode : tan 0.1234 0.1228
Check:
Degree mode : tan 7.0348 0.1234
Radian mode : tan 0.1228 0.1234
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