4.3
Fundamental Trig
Identities and Right
Triangle Trig
Applications
2015
Trig Identities Sheet
Handout
Trigonometric Identities are trigonometric
equations that hold for all values of the variables.
Example: sin  = cos(90  ), for 0 <  < 90
Note that  and 90  are complementary
angles.
Side a is opposite θ and also
adjacent to 90○– θ .
hyp
θ
a
a
sin  =
and cos (90  ) =
.
hyp
hyp
So, sin  = cos (90  ).
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90○– θ a
b
11
Fundamental Trigonometric Identities for 0 <  < 90.
Cofunction Identities
sin  = cos(90  )
tan  = cot(90  )
sec  = csc(90  )
cos  = sin(90  )
cot  = tan(90  )
csc  = sec(90  )
Reciprocal Identities
sin  = 1/csc 
cot  = 1/tan 
cos  = 1/sec 
sec  = 1/cos 
tan  = 1/cot 
csc  = 1/sin 
Quotient Identities
tan  = sin  /cos 
cot  = cos  /sin 
Pythagorean Identities
sin2  + cos2  = 1
tan2  + 1 = sec2 
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cot2  + 1 = csc2 
12
Negative Angle Identities
Remember:
if f(-t) = f(t) the function is even
if f(-t) = - f(t) the function is odd
The cosine and secant functions are EVEN.
cos(-t)=cos t
sec(-t)=sec t
(0,1) y
(–1, 0)
(1, 0)
(0,–1)
The sine, cosecant, tangent, and cotangent functions are
ODD.
sin(-t)= -sin t
csc(-t)= -csc t
tan(-t)= -tan t
cot(-t)= -cot t
x
Example: Given sec  = 4, find the values of the
other five trigonometric functions of  .
Draw a right triangle with an angle  such
4
4
hyp
that 4 = sec  =
= .
adj 1
Use the Pythagorean Theorem to solve
for the third side of the triangle.
15
4
cos  = 1
4
tan  = 15 = 1 5
1
sin  =
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15
θ
1
1 = 4
sin 
15
1
sec  =
=4
cos 
1
cot  =
15
csc  =
14
Example:
Given sin  = 2/5, find the values of the other five
trigonometric functions of  .
θ
tan  =
cot  =
cos  =
sec =
csc  =
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15
Use trigonometric identities to find the
indicated trigonometric functions.
 a  sin
2
322  cos 322
2
(c) csc 30
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 b  cot 60
 d  sec 60
16
Use trigonometric identities to transform one
side of the equation into the other.
a. cos sec  1
c. csc tan  sec
b. (sec   tan  )(sec   tan  )  1
tan   cot 
 csc 2 
d.
tan 
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17
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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18
Homework
4.3 p 274 5-15 odd, 27-43 odd, 47-55 odd
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20
Degrees, Minutes, and Seconds
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.
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21
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 152.65 in degrees, minutes, and seconds
15239
Express 522822 in decimal degrees
52.47277