Download tan θ

definitions of trigonometric functions with (x, y) on their terminal sides:
r
y
csc θ =
sin θ = y
r
cos θ =
tan θ =
x
r
sec θ =
y
x
cot θ =
r
x
r
x
y
(x, y)
y
x
where r = √x2+y2
Reciprocal Identities
sin θ =
cos θ =
tan θ =
1
csc θ
1
sec θ
1
cot θ
sec θ =
cot θ =
csc θ =
Pythagorean Identities
sin2θ + cos2θ = 1
1 + cot2θ = csc2θ
tan2 θ
sin2θ + 1 = sec2θ
1
sin θ
1
cos θ
1
tan θ
Ratio Identities
tan θ =
sin θ
cos θ
cot θ = cos θ
sin θ
Right Triangle Trigonometry ­­ almost the same as what we just did, but the sides are labeled differently. r = the hypotenuse = hyp
x and y are the opposite and adjacent sides, opp and adj, with order depending upon where you put the angle.
opp
hyp
sin θ = hyp
csc θ = opp
cos θ =
adj
hyp
sec θ =
tan θ =
opp
adj
cot θ = adj
opp
hyp
opp
α
β
hyp
adj
adj
hyp
adj
where θ is either angle, α or β
opp
EXAMPLE:
If we have the following triangle, find the values of all six trigonometric functions for both α and β.
β
13
12
α
5
Let's find the exact values of the six trigonometric functions for 30o and 60o using our special
triangle with a base side of 1. sin 30o =
csc 30o =
cos 30o =
sec 30o =
tan 30o =
cot 30o =
sin 60o =
csc 60o =
cos 60o =
sec 60o =
tan 60o =
cot 60o =
1. If α + β = 90o, how do cos α and sin β relate to each other?
Cofunctions:
sin θ = cos (90o ­ θ)
cos θ = sin (90o ­ θ)
tan θ = cot (90o ­ θ)
cot θ = tan (90o ­ θ)
2. How do cos β and sin α relate to each other?
Why do you think this happened? Will it always happen?
sec θ = csc (90o ­ θ)
csc θ = sec (90o­ θ)
Section 2.2
Angles can be divided into smaller pieces.
1 degree = 60 minutes and 1 minute = 60 seconds
1o = 60' 1' = 60"
Adding and subtracting in degrees, minutes, and seconds:
Add 42o 12' 27" to 5o 50' 3"
Subtract 18o 43' from 25o 5'
Changing decimal degrees to degrees, minutes, seconds:
Example: change 25o 4' 15" to decimal degrees.
Example: change 20.3o to degrees, minutes, seconds
Note: you may use your calculator to do these changes if you wish. Degrees, minutes, seconds is abbreviated DMS on most calculators.
Using your calculator to find trigonometric functions:
VIP: Make sure your mode is in degrees when the angle is in degrees.
If the angle is in decimal degrees, you can find sin, cos, or tan directly.
sin 55o =
cos 48.2o =
tan 78.72o =
To find csc, sec, or cot, use the reciprocal of sin, cos, and tan.
csc 48.1o = 1/sin 48.1o =
sec 19.4o =
cot 8.75o =
If the angle is in degrees, minutes, seconds, you can use them directly, finding the appropriate symbols (often under math and angles or directly on the keyboard) on your calculator or you can convert them to decimal degrees first.
Find sin 32o10'48".
Now for something more exciting! When an angle is between 0o and 90o,
what do you know about the signs of the trigonometric functions of that
angle?
If we have the value of a trigonometric function of an angle
between 0o and 90o we can use our calculators to find the
angle using the sin-1, cos-1, and tan-1 buttons.
If sin A = 0.3971, then A = sin-1 0.3971 =
If cot B = 0.4327, then we must find tan B first.
tan B = 1/0.4327 and B = tan-1(1/0.4327) =
Find the angle between 0o and 90o for each of the following.
1. cos C = 0.5490
2. tan A = 0.6273
3. csc B = 1.4293
4. sec D = 1.0801
Now, look at problems 85­88 on p. 69. 85. What happens when you try to find A for sin A = 1.234 on your calculator? Why does it happen?
86. What happens when you try to find B for sin B = 4.321 on your calculator? Why does this happen?
87. What happens when you try to find tan 90o on your calculator? Why does this happen?
88. What happens when you try to find cot 90o on your calculator? Why does this happen?