4.3 fundamental trig identities and right triangle trig applications 2015 trig identities sheet...
DESCRIPTION
3 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: the side opposite the acute angle , the side adjacent to the acute angle , and the hypotenuse of the right triangle. The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp adj hyp θ Trigonometric Functions sin = cos = tan = csc = sec = cot = opp hyp adj hyp adj opp adjTRANSCRIPT
4.3Fundamental Trig
Identities and Right Triangle Trig Applications
2015Trig Identities Sheet
Handout
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Trigonometric Identities are trigonometric equations that hold for all values of the variables.
Example: sin = cos(90 ), for 0 < < 90Note that and 90 are complementary angles.
Side a is opposite θ and also adjacent to 90○– θ .
ahyp
bθ
90○– θ
sin = and cos (90 ) = .
So, sin = cos (90 ).
ahyp
ahyp
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Fundamental Trigonometric Identities for 0 < < 90.Cofunction Identities
sin = cos(90 ) cos = sin(90 )tan = cot(90 ) cot = tan(90 )sec = csc(90 ) csc = sec(90 )
Reciprocal Identities
sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin
Quotient Identities
tan = sin /cos cot = cos /sin
Pythagorean Identities
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2
Negative Angle IdentitiesRemember:
if f(-t) = f(t) the function is evenif f(-t) = - f(t) the function is odd
The cosine and secant functions are EVEN.cos(-t)=cos t sec(-t)=sec t
The sine, cosecant, tangent, and cotangent functions are ODD.
sin(-t)= -sin t csc(-t)= -csc ttan(-t)= -tan t cot(-t)= -cot t
(1, 0)(–1, 0)
(0,–1)
(0,1)
x
y
x
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Example: Given sec = 4, find the values of the other five trigonometric functions of .
Use the Pythagorean Theorem to solve for the third side of the triangle.
tan = = cot =115
151
15
sin = csc = =415
154
sin1
cos = sec = = 4 41
cos1
1 5
θ
4
1
Draw a right triangle with an angle such that 4 = sec = = .
adjhyp
14
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Example: Given sin = 2/5, find the values of the other five trigonometric functions of .
tan = cot =
cos = sec =
csc =
θ
Use trigonometric identities to find the indicated trigonometric functions.
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2 2sin 322 cos 322 cot 60
( )csc30 sec60
a b
c d
Use trigonometric identities to transform one side of the equation into the other.
a. b.
c. d.
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cos sec 1 (sec tan )(sec tan ) 1
csc tan sec tan cot
tancsc
2
Applying Trig
You are 200 yards from a river. Rather than walking directly to the river, you walk 400 yards, diagonally, along a straight path to the rivers edge. Find the acute angle between this path and the river’s edge.
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Homework
4.3 p 274 5-15 odd, 27-43 odd, 47-55 odd
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There is another way to state the size of an angle, one that subdivides a degree into smaller pieces.
In a full circle there are 360 degrees. Each degree can be divided into 60 parts, each part being 1/60 of a degree. These parts are called minutes.
Each minute can divided into 60 parts, each part being 1/60 of a minute. These parts are called seconds.
Degrees, Minutes, and Seconds
Degrees, Minutes, and Seconds Conversions
•To convert decimal degrees into DMS, multiply decimal degrees by 60
•To convert from DMS to decimal degrees, divide minutes by 60, seconds by 3600
•OR use the Angle feature of your calculator
•Examples
Express 52 28 22 in decimal degrees
Express 152.65 in degrees, minutes, and seconds
152 39
52.47277