trigonometry. right triangles non-right triangles 1. trig functions: sin, cos, tan, csc, sec, cot 2....
TRANSCRIPT
![Page 1: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/1.jpg)
Trigonometry
![Page 2: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/2.jpg)
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
![Page 3: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/3.jpg)
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
![Page 4: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/4.jpg)
Confused?
hypotenuse
opposite
adjacent
![Page 5: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/5.jpg)
hypotenuseopposite
adjacent
![Page 6: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/6.jpg)
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.
![Page 7: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/7.jpg)
![Page 8: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/8.jpg)
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.”
![Page 9: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/9.jpg)
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
![Page 10: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/10.jpg)
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
![Page 11: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/11.jpg)
Special Triangles: 30-60-90 and 45-45-90
30˚
60˚
45˚
45˚1
1
1
2 3 2
![Page 12: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/12.jpg)
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
![Page 13: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/13.jpg)
Find the exact values of x and y.
60˚x
8 y
![Page 14: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/14.jpg)
Find the values of x and y.
35˚y
x 16
![Page 15: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/15.jpg)
This is 1 unit long.
180˚ = π radians
360˚ = 2π radians
Hence the name:
The UNIT CIRCLE
90˚ = π radians 2
![Page 16: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/16.jpg)
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!
![Page 17: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/17.jpg)
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!!!
![Page 18: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/18.jpg)
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˚
![Page 19: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/19.jpg)
●
●●
●(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)
![Page 20: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/20.jpg)
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 )
![Page 21: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/21.jpg)
![Page 22: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/22.jpg)
What are the coordinates
of ●
θ
( the length of the pink line, the length of the red line)
( cos θ, sin θ )
![Page 23: Trigonometry. Right Triangles Non-Right Triangles 1. Trig Functions: Sin, Cos, Tan, Csc, Sec, Cot 2. a 2 + b 2 = c 2 3. Radian Measure of angles 4. Unit](https://reader036.vdocument.in/reader036/viewer/2022062309/5697bfbe1a28abf838ca27b9/html5/thumbnails/23.jpg)
The UNIT CIRCLE
●
●●
●(4, 12)(4, 12)
adjacent
opposite
hypo
tenu
se
radi
us