Trigonometry Recall: hypotenuse opposite adjacent 1
There are 3 other ratios: the reciprocals of sine, cosine and tangent. Secant: Cosecant: (cosec θ) Cotangent: 2
Example: Determine the value of x. a) b) c) 3
Radians The radian (rad) is a unit of angle measure. When a wheel completes a turn... 1) how many degrees has it moved? 2) how far has the rim of the wheel travelled? 3) How many times is the radius contained in the circumference? 4
r 1 rad r A B r When the distance covered by the circumference is equal to the length of the radius (i.e., ), the measure of the central angle is equal to 1 rad. There are radians in a circle. radians 5
We can convert between radians and degrees using the proportion or 6
Example: What is... a) in radians? b) rad in degrees? c) rad in degrees? d) in radians? 7
Arc Length Recall: central angle arc length 360 circumference Replacing with radians: Arc Length: L Note: must be in radians. 8
Examples: 1. Determine the length of the arc, given and. or 9
2. Determine the diameter of the circle if and. 10
3. Determine the measure of the central angle, if and. 11
4. Determine the length of the arc (using ) if and. or 12
Trigonometric Angles Terminal side y Initial side A trigonometric angle has its vertex at the origin, an initial side on the x -axis and a x terminal side obtained by a rotation. If the rotation is counterclockwise, then the angle is positive. If the rotation is clockwise, then the angle is negative. 13
Determine the measure of the trigonometric angle shown below, in degrees and radians. y 40 50 60 130 120 70 110 80 100 90 90 100 80 70 110 60 120 50 130 140 20 30 160 150 140 0 40 30 20 150 160 x 10 0 170 180 10 170 0 180 14
Represent the trigonometric angle rad. y x 40 50 60 130 120 70 110 80 100 90 90 100 80 70 110 60 120 50 130 140 20 30 160 150 140 0 40 30 20 150 160 10 170 10 170 0 180 0 180 15
In which quadrant would you find the following trigonometric angles? a) d) b) e) c) 16
Trigonometric Circle The trigonometric circle is a unit circle centred at the origin. y Any point on the circle is called a trigonometric point. 1 1 0 1 x 1 Every trigonometric angle has a corresponding trigonometric point. 17
Known trig points: y 1 x 1 0 1 1 18
y 1 1 0 1 x 1 Notice: and 19
and y 1 x 1 0 1 1 The trig point for any trigonometric angle is. 20
Example: a) P(50 ) = b) 21
There are other "special angles" for which we can determine the exact coordinates of their trigonometric points. y 1 1 0 1 x 1 22
1 or rad 23
2 24
25
y 1 x 1 0 1 By reflecting these points around the axes, we can determine 12 more points with exact coordinates. 1 26
27
This circle with all of the exact trigonometric points is known as the trigonometric circle. 28
Example: Determine the exact values for the following points. a) b) 29
c) d) e) f) g) 30
If... and, then... =? 31
Example: Determine the exact values of... a) b) c) d) 32
Determine the sign of, and in each of the four quadrants. y x 33
Sum or Difference of Two Angles There are formulae that allow us to determine the sine, cosine and tangent of the sum or difference of two angles. Sine Cosine Tangent Sum Difference 34
We can use these rules to determine the exact values of some angles not found on the trig circle. Examples: Determine the exact values of... 1. 2. 35
3. 4. 5. 36
We can also use these rules to prove some statements. Examples: Show that... 1. 2. 37
3. 38
Find an expression that corresponds to... 1. 2. These expressions are also rules and are known as the double angle formulas. 39
Trigonometric Identities An identity is a statement that is true for all possible values of the variable. 1) 2) 3) 4) 5) 6) 40
7) 8) 41
Pythagorean Identities 9) i) ii) 42
We can take this identity and eliminate (in turn) the and the to create two new identities. 10) i) ii) 43
11) i) ii) We use identities to simplify statements or prove that statements are true. 44
Examples: Simplify each of the following expressions. a) b) c) d) 45
Examples: Prove each of the following identities. a) Simplify the more "complicated " side 46
b) 47
More trigonometric identities: a) b) c) 48
d) e) f) 49
Solving Trigonometric Equations To solve a trigonometric equation means to find the value of the angle. We may have to use simplifying techniques. Example 50
Examples: Solve each equation. a) b) c) d) 51
Solve the following trigonometric equations. a) b) c) d) e) 52