MHF4U – Unit 4: Trigonometry – Radians
REVIEW
IMPORTANT THINGS TO REMEMBER:
Reciprocal Trig Ratios:
Special Triangles:
𝑐𝑠𝑐𝜃 =
1
𝑠𝑖𝑛𝜃
=
𝐻
𝑠𝑒𝑐𝜃 =
𝑂
1
𝑐𝑜𝑠𝜃
𝝅
𝝅
𝟒
𝟔
=
𝐻
45
1
=
𝑡𝑎𝑛𝜃
𝐴
𝑂
Remember: 𝝅 = 𝟏𝟖𝟎°
√𝟐
1
𝑐𝑜𝑡𝜃 =
𝐴
2
so
𝟏𝟖𝟎°
𝟑
=
𝝅
𝟑
= 𝟔𝟎°...
√𝟑
1
45 ̊
60 ̊
1
CAST rule: defines where Cosine, All, Sine, Tan lines are positive; if the line falls within a quadrant, that expression is
positive to the corresponding letter. Label from bottom right, up, left, and down to bottom left. Everything
positive in the A (all) quadrant.
RAA: Related Acute Angle: the smallest angle, under 90 degrees, that connects the line to the x-axis
CTA: Co-Terminal Angle: the smallest angle between 0 degrees and the terminal arm. EX: 900 degrees has CTA = 180
𝝅
Degrees  Radians: multiply the degrees by ( )
𝟏𝟖𝟎
Radians  Degrees: 𝜋 = 180°
2𝜋
Transformations: root expression – 𝑦 = 𝒂 sin k(𝜃 − 𝑑) + 𝑙 where a= amplitude, k=
, d=phase shift, l = vert. trans.
period
AROC: 𝑦 =
𝑓(𝑏)−𝑓(𝑎)
IROC: 𝑦 =
𝑏−𝑎
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
DEGREES
1. Find the six trig ratios of angle θ whose terminal arm passes through (-3,5).
2. Find the exact value of 𝑠𝑖𝑛2 45 + 25𝑠𝑖𝑛30𝑠𝑒𝑐30.
Homework
3. Find the exact value of each using the CAST rule and the RAA rule.
a) cos150 ̊
b) tan220 ̊
CONVERSION
4. Convert the following to radians.
a) 45 degrees
b) 90 degrees
c) 225 degrees
5. Convert the following to degrees.
𝜋
2𝜋
a) rad
b) rad
c) 0.63 rad
3
5
RADIANS
6. Complete the number line.
Radian Measure: pg 321 1-6
Transformations: pg 343 1,4,6,7,8
Inverse Functions: pg 353 2-7
Applications: pg 360 6-10
AROC & IROC: pg 370 2,6,7,9,12
Review: pg 376 1-6, 8-19
𝜋
2𝜋
7𝜋
3𝜋
3
3
6
2
7. Find the RAA, CAST and CTA of the following.
9𝜋
19𝜋
−7𝜋
a) 𝑠𝑖𝑛
b) 𝑐𝑜𝑠
c) 𝑐𝑜𝑡
4
6
4
8. Determine the exact value of each of the following. Draw a diagram for each.
5𝜋
11𝜋
−3𝜋
a) 𝑡𝑎𝑛
b) 𝑐𝑠𝑐
c) 𝑐𝑜𝑡
6
3
4
9. Complete the following table of transformations in radians.
amplitude
k value
𝒚 = 𝟑𝒔𝒊𝒏(𝟐𝜽 + 𝝅) − 𝟏
𝝅
𝒚 = 𝟓𝒄𝒐𝒔 (−𝟐𝒙 + ) − 𝟐
𝟑
𝒚 = 𝟎. 𝟓𝒄𝒐𝒔(𝟒𝒙)
𝝅
𝒅(𝒕) = −𝟐𝒔𝒊𝒏𝟐 (𝒕 + )
𝟒
period
phase shift
vert. trans.
10. Draw a graph for each of the equations in 9. Include the original function and mapping.
11. Determine the equations of the following transformed functions (the original function is darker). Use the table
below to assist you.
a)
𝝅
X-scale: , Y-scale: 1
𝟐
AMP:
PER:
P.S.:
V.T.:
b)
𝝅
X-scale: , Y-scale: 1
𝟐
AMP:
PER:
P.S.:
V.T.:
c)
𝝅
X-scale: , Y-scale: 1
𝟐
AMP:
PER:
P.S.:
V.T.:
12. A spring with a weight on the end is suspended from the ceiling of a room and it bounces up and down with an
amplitude of 60 cm making 15 complete cycles every minute. Let y denote the height of the weight above the
equilibrium position.
a) Draw a graph of y against t assuming that at t=0 the weight is at the equilibrium point y=0 and moving
upwards. Show 2 full cycles.
b) Find an equation for your curve in radian mode.
c) Locate on your graph all times at which the weight has height y=30 cm, and use your formula of (b) to
calculate these exactly.
13. Study the following sine graph. The x-scale is in intervals of 2. The y-scale is in intervals of 50.
Determine the amplitude and period of the sine curve in radian mode.
1
𝜋
2
4
14. Find the average rate of change (AROC) of 𝑦 = −5 𝑠𝑖𝑛 ( 𝑥) − 9 for the interval ≤ 𝑥 ≤ 𝜋.
𝜋
15. A ship that is docked in a harbour rises and falls with the waves. The function ℎ(𝑡) = sin ( 𝑡)models the
5
vertical movement of the ship, h in metres, at t seconds.
a) Determine the average rate of change (AROC) in the height of the ship over the first 5 seconds.
b) Estimate the instantaneous rate of change in the heights of the ship at t=6.
GRAPH IT!
Changing Mode to Radians: press MODE, cursor to “Radians”, press ENTER
Some questions in this review were borrowed from Nelson Advanced Functions© text. Shhh!