trigonometry. right triangles non-right triangles 1. trig functions: sin, cos, tan, csc, sec, cot 2....
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Trigonometry
Trigonometry
Right Triangles Non-Right Triangles
1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot2. a2 + b2 = c2
3. Radian Measure of angles4. Unit circle5. Inverse trig functions
1. Exact values
3. Changing units.
5. Calculator work
1. Law of Sines2. Law of Cosines
: AAS, ASA, SSA: SAS, SSS
Right Triangles“naming the sides of the triangle.”
What we call the
legs of the triangle
depend on the non-
right angle given.
hypotenuse
opposite
adjacentopposite
adjacentThis is important because all of the trig functions are ratios that are defined by
the lengths of these sides. For example: sine of an angle is the ratio of
the length of the side opposite the angle divided by the length of the
hypotenuse.
Sin θ = hypopp
Confused?
hypotenuse
opposite
adjacent
hypotenuseopposite
adjacent
Trig FunctionsThere are 6 trig functions we
must be able to use. We must memorize their EXACT values in both radical and radian form. Remember: trig functions are
the result of ratios of the lengths of sides of a right triangle.
Trig FunctionsThere are 3 main trig functions and the 3 that are reciprocals of
the first three. The main ones are: Sine, Cosine
and Tangent.sin θ = opp hyp
cos θ = adj hyp
tan θ = opp adj
The reciprocals are: Cosecant, Secant and Cotangent.
csc θ = hyp opp
sec θ = hyp adj
cot θ = adj opp
Basically, to find the trig
relationship of any angle on a
right triangle, all we need to do is
measure the appropriate sides of that triangle.
This is called “evaluating the trig
functions of an angle θ.”
hypotenuseopposite
adjacent
Evaluate the six trig functions of the angle θ.
θ
sin θ = opp hyp
3
4
5
sin θ = 3 5
cos θ = adj hyp cos θ = 4 5
tan θ = opp adjtan θ = 3 4
csc θ = 5 3
sec θ = 5 4
cot θ = 4 3
hypotenuse
opposite
adjacent
We can work backwards as well. If they give us the ratio, we can find the other trig
functions.θ
sin θ = opp hyp
Given: sin θ = 5 6
cos θ = 6
tan θ = 5
sec θ = 6
csc θ = 6 5
cot θ = 5
5
6a2 + b2 = c2
11
1111
11
11
sec θ = 6 11
11
tan θ = 5 11
11
Special Triangles: 30-60-90 and 45-45-90
30˚
60˚
45˚
45˚1
1
1
2 3 2
30˚
60˚
45˚
θ sin θ cos θ
tan θ
csc θ
sec θ
tan θ
30˚
60˚
45˚
θ sin θ cos θ
tan θ
csc θ
sec θ
tan θ
Find the exact values of x and y.
60˚x
8 y
Find the values of x and y.
35˚y
x 16
This is 1 unit long.
180˚ = π radians
360˚ = 2π radians
Hence the name:
The UNIT CIRCLE
90˚ = π radians 2
Since 180 ˚ = 1π radians we can us this as our conversion factor.
In other words to change
degrees into radian we multiply by π
180˚
To change
radians into degrees we multiply by
π
180˚
Hint: What we “want” is always in the numerator. If we want our final answer in degrees then 180 ˚ is on
top. If we want radians then π radians in on top!
Convert 230˚ to radians.
Since we want radians we multiply by π/18
(radians in the numerator.
230˚● π = 230π180˚ 180
Which reduces to 23π 18
NO MIXED FRACTIONS!!!
Convert π to degrees12
Since we want degrees we multiply by 180/π
(degrees in the numerator.)Notice the π’s
cancel!
180
12
Reduces to 15˚
●
●●
●(4, 12)(4, 12)
adjacent
opposite
hypo
tenu
se
radi
us
This leads us to believe that there must be a connection between
sin, cos and the coordinates (x, y)
The UNIT CIRCLE
Remember the unit circle has a radius of 1 unit.
●
So to find the coordinates of this point we can use the sin and cos if we know what the measure of the
angle formed by the radius and the x axis is..
θ
( the length of the pink line, the length of the red line)
BUT WAIT! That’s what cos and sin are defined as!
sinθ = length of side opposite length of hypotenuse
cosθ = length of side adjacent length of hypotenuse
AND WE KNOW THAT THE RADIUS IN A UNIT CIRCLE IS 1 so that means:
sinθ = length of side opposite
cosθ = length of side adjacent
( cos θ, sin θ )
What are the coordinates
of ●
θ
( the length of the pink line, the length of the red line)
( cos θ, sin θ )
The UNIT CIRCLE
●
●●
●(4, 12)(4, 12)
adjacent
opposite
hypo
tenu
se
radi
us
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