Angles & Circular Functions
Angles & Circular Functions Trig Functions of Special Angles From Geometry ... remember the ratios for 30-60-90 and 45-45-90 triangles?
2 ! leg Now let’s put these triangles onto a unit circle, and we’ll be able to evaluate the trig functions at angles that are 30°, 60° and 45°. Remember, on a unit circle, the hypotenuse, r = 1, sin ! = y and cos! = x .
Sine and Cosine of 30°, 60° and 45°
Sine and Cosine of 0° and 90°
All Six Trig Functions Once you know the sine and cosine values for an angle, you can find any other trig value for that angle. These formulae come from the definitions of the trig functions on the coordinate plane.
y r x Since cos! = r y Since sin ! = r y Since sin ! = r
Since sin ! =
r 1 , we can say that csc ! = . y sin ! r 1 and sec ! = , we can say that sec ! = . x cos! x y sin ! and cos! = and tan ! = , we can say that tan ! = . r x cos! x x cos! and cos! = and cot ! = , we can say that cot ! = . y r sin ! and csc ! =
Using the sine and cosine values we have found and the rules about the other four trig functions, we’ll put what we’ve learned into a table: Trig Functions of Special Angles
Angle, !
sin !
cos!
tan !
csc !
sec !
cot !
0
0
1
0
und.
1
und.
! 6
1 2
3 2
3 3
2
2 3 3
3
! 4
2 2
1
2
2
1
! 3 ! 2
3 2
2 2 1 2
3
2 3 3
2
3 3
1
0
und.
1
und.
0
You will need to memorize the information in this table, here are some tips that will make it easier to memorize the information. 1. Look at the sin ! column, notice that the values increase as the angles increase. Memorize the data and you will be able to repeat this column in the proper order. 2. Now look at the cos! column, it is the reverse of the sin ! column! sin ! 3. The tan ! column is the quotient of the sin ! and cos! columns: . cos! 4. The csc ! column is the reciprocal of the sin ! column. 5. The sec ! column is the reciprocal of the cos! column. 6.The cot ! column is the reciprocal AND the reverse of the tan ! column.
Trig Functions of Angles Outside of Quadrant I: All of the trig values that we have looked at so far have been for angles in the first quadrant. ! You will also need to evaluate trig functions of angles larger than 90° or radians and negative 2 angles as well. To evaluate trig functions of other angles, you will need to find the reference angle. Example:
Evaluate sin
5! . 6
5! 6
Sketch the angle. Its terminal side is in Q2.
Draw the right triangle formed by the reference angle. The reference angle =
5! 6
! . 6
5" " will have the same values as they do for ! = , 6 6 except for the sign of the function since x, y and r are not all positive any more. An angle in Quadrant 2 has a negative x, a positive y and a positive r (r is always positive).
The trig functions for ! =
y 5! and since r and y are both positive in Quadrant 2, sin will be r 6 5! ! positive. Also, sin will have the same magnitude as its reference angle, sin . 6 6 5! 1 Thus sin = . 6 2
Since sin ! =
Here is a quick way to determine the sign of a trig function. Wherever x is positive, COSINE (and secant) are positive, wherever y is positive, SINE (and cosecant) are positive. Wherever x and y have the same sign, TANGENT (and cotangent) are positive. In all other cases, the trig ratios are negative.
SIN is + COS is TAN is -
SIN is + COS is + TAN is +
SIN is COS is TAN is +
SIN is COS is + TAN is -
A trick to remember the signs of the trig functions:
All Star Trig Class!
To evaluate a trig function of an angle outside of quadrant I, 1. Find the reference angle. 2. Evaluate the trig function of the reference angle. 3. Determine the sign of the trig function based on the quadrant that the original angle lies in (All Star Trig Class). More Examples: 1. Evaluate sec
5! . 4
5! , lies in Q3, where cos ! and sec ! are negative. 4 ! ! The reference angle is and sec = 2 . 4 4 5! The answer is sec =" 2. 4
The terminal side of the angle,
2. Evaluate tan 495°. The angle 495° is larger than 360°. After the angle wraps around the circle once it continues to 135°, which lies in Q2, where tan ! is negative. The reference angle is 45° and tan 45° = 1. The answer is tan 495° = -1.
# "& 3. Evaluate cos % ! ( . $ 6' Remember, negative angles open clockwise from the initial side! " ! ! lies in Q4, where cos ! is positive. Its reference angle is . 6 6 3 # "& cos % ! ( = $ 6' 2
Try these!
# "& 1) Evaluate sin % ! ( $ 6' 3! 3) Evaluate sec 2 11! 5) Evaluate cos 3
" 5! % 2) Evaluate tan $ ' # 6 & 4) Evaluate cot ( !3" )
# 11" & 6) Evaluate csc % ! $ 3 ('
2 ! leg Now let’s put these triangles onto a unit circle, and we’ll be able to evaluate the trig functions at angles that are 30°, 60° and 45°. Remember, on a unit circle, the hypotenuse, r = 1, sin ! = y and cos! = x .
Sine and Cosine of 30°, 60° and 45°
Sine and Cosine of 0° and 90°
All Six Trig Functions Once you know the sine and cosine values for an angle, you can find any other trig value for that angle. These formulae come from the definitions of the trig functions on the coordinate plane.
y r x Since cos! = r y Since sin ! = r y Since sin ! = r
Since sin ! =
r 1 , we can say that csc ! = . y sin ! r 1 and sec ! = , we can say that sec ! = . x cos! x y sin ! and cos! = and tan ! = , we can say that tan ! = . r x cos! x x cos! and cos! = and cot ! = , we can say that cot ! = . y r sin ! and csc ! =
Using the sine and cosine values we have found and the rules about the other four trig functions, we’ll put what we’ve learned into a table: Trig Functions of Special Angles
Angle, !
sin !
cos!
tan !
csc !
sec !
cot !
0
0
1
0
und.
1
und.
! 6
1 2
3 2
3 3
2
2 3 3
3
! 4
2 2
1
2
2
1
! 3 ! 2
3 2
2 2 1 2
3
2 3 3
2
3 3
1
0
und.
1
und.
0
You will need to memorize the information in this table, here are some tips that will make it easier to memorize the information. 1. Look at the sin ! column, notice that the values increase as the angles increase. Memorize the data and you will be able to repeat this column in the proper order. 2. Now look at the cos! column, it is the reverse of the sin ! column! sin ! 3. The tan ! column is the quotient of the sin ! and cos! columns: . cos! 4. The csc ! column is the reciprocal of the sin ! column. 5. The sec ! column is the reciprocal of the cos! column. 6.The cot ! column is the reciprocal AND the reverse of the tan ! column.
Trig Functions of Angles Outside of Quadrant I: All of the trig values that we have looked at so far have been for angles in the first quadrant. ! You will also need to evaluate trig functions of angles larger than 90° or radians and negative 2 angles as well. To evaluate trig functions of other angles, you will need to find the reference angle. Example:
Evaluate sin
5! . 6
5! 6
Sketch the angle. Its terminal side is in Q2.
Draw the right triangle formed by the reference angle. The reference angle =
5! 6
! . 6
5" " will have the same values as they do for ! = , 6 6 except for the sign of the function since x, y and r are not all positive any more. An angle in Quadrant 2 has a negative x, a positive y and a positive r (r is always positive).
The trig functions for ! =
y 5! and since r and y are both positive in Quadrant 2, sin will be r 6 5! ! positive. Also, sin will have the same magnitude as its reference angle, sin . 6 6 5! 1 Thus sin = . 6 2
Since sin ! =
Here is a quick way to determine the sign of a trig function. Wherever x is positive, COSINE (and secant) are positive, wherever y is positive, SINE (and cosecant) are positive. Wherever x and y have the same sign, TANGENT (and cotangent) are positive. In all other cases, the trig ratios are negative.
SIN is + COS is TAN is -
SIN is + COS is + TAN is +
SIN is COS is TAN is +
SIN is COS is + TAN is -
A trick to remember the signs of the trig functions:
All Star Trig Class!
To evaluate a trig function of an angle outside of quadrant I, 1. Find the reference angle. 2. Evaluate the trig function of the reference angle. 3. Determine the sign of the trig function based on the quadrant that the original angle lies in (All Star Trig Class). More Examples: 1. Evaluate sec
5! . 4
5! , lies in Q3, where cos ! and sec ! are negative. 4 ! ! The reference angle is and sec = 2 . 4 4 5! The answer is sec =" 2. 4
The terminal side of the angle,
2. Evaluate tan 495°. The angle 495° is larger than 360°. After the angle wraps around the circle once it continues to 135°, which lies in Q2, where tan ! is negative. The reference angle is 45° and tan 45° = 1. The answer is tan 495° = -1.
# "& 3. Evaluate cos % ! ( . $ 6' Remember, negative angles open clockwise from the initial side! " ! ! lies in Q4, where cos ! is positive. Its reference angle is . 6 6 3 # "& cos % ! ( = $ 6' 2
Try these!
# "& 1) Evaluate sin % ! ( $ 6' 3! 3) Evaluate sec 2 11! 5) Evaluate cos 3
" 5! % 2) Evaluate tan $ ' # 6 & 4) Evaluate cot ( !3" )
# 11" & 6) Evaluate csc % ! $ 3 ('
