Trig Functions Review
(Including Trig Quiz Solutions)
MHF4UI
Friday November 16th, 2012
Convert the following angles to Radians.
Provide Exact Answers
𝜋
1° =
180
270𝜋
270° =
180
3𝜋
=
2
𝜋
1° =
180
900𝜋
900° =
180
= 5𝜋
Convert the following angles to Degrees. Round
to the nearest degree.
180
1𝑟𝑎𝑑 = (
)°
𝜋
8
8 180
=( •
)°
𝜋
𝜋 𝜋
≈ 146°
180
1𝑟𝑎𝑑 = (
)°
𝜋
5.75 • 180
5.75 = (
)°
𝜋
≈ 329°
Question 3: Finding the Arc Length
𝑎
𝜃=
𝑟
𝑎 = 𝑟𝜃
11𝜋
𝑎 = 15
6
𝑎 ≈ 86.39
Therefore he travelled a distance of about 86.39 metres.
11𝜋
𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝑝𝑒𝑟 ℎ𝑜𝑢𝑟
6
11𝜋
𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝑝𝑒𝑟 𝑚𝑖𝑛
6 • 60
Therefore his angular velocity is
11𝜋
360
per minute.
Question 6: Solve the Equation
5 sin 𝑥 − 6 sin 𝑥 + 5 = 6
− sin 𝑥 = 1
sin 𝑥 = −1
Why must we have sec 𝜃 ≥ 1 𝑜𝑟 sec 𝜃 ≤ −1
sec 𝜃 =
ℎ𝑦𝑝
𝑎𝑑𝑗
In a right angled triangle, the absolute value of the length of the hyp will always be
greater than or equal to absolute value of the length of the adj
ℎ𝑦𝑝
≥1
𝑎𝑑𝑗
Therefore sec 𝜃 ≥ 1 𝑜𝑟 sec 𝜃 ≤ −1
What is a Radian?
Much like a degree, a Radian is a measurement of an angle.
The radian measure of an angle ,ϴ, is defined as the length, a, of
the arc that “subtends” the angle divided by the radius of the arc ,r
𝑎
𝜃=
𝑟
Radian Relationship to Degrees
180
1 𝑟𝑎𝑑𝑖𝑎𝑛 = (
)°
𝜋
𝜋
1° =
𝑟𝑎𝑑𝑖𝑎𝑛𝑠
180
Word Problems
𝑆𝑜𝑙𝑣𝑖𝑛𝑔 𝑓𝑜𝑟 𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ, 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑟 𝜃
𝐹𝑖𝑛𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑎 𝑟𝑜𝑡𝑎𝑡𝑖𝑛𝑔 𝑜𝑏𝑗𝑒𝑐𝑡
Trig Ratios
Within a right angled triangle we will have that:
𝑜𝑝𝑝
𝑠𝑖𝑛𝜃 =
ℎ𝑦𝑝
𝑎𝑑𝑗
𝑐𝑜𝑠𝜃 =
ℎ𝑦𝑝
𝑜𝑝𝑝
𝑡𝑎𝑛𝜃 =
𝑎𝑑𝑗
ℎ𝑦𝑝
𝑐𝑠𝑐𝜃 =
𝑜𝑝𝑝
ℎ𝑦𝑝
𝑠𝑒𝑐𝜃 =
𝑎𝑑𝑗
𝑎𝑑𝑗
𝑐𝑜𝑡𝜃 =
𝑜𝑝𝑝
Finding Trig Ratios
We found Trig Ratios using our Related Acute angles of:
𝜋 𝜋 𝜋
, ,
4 3 6
We found trig ratios by drawing our angles in standard position.
We found trig ratios when our terminal arm lies on the x or y axis.
We also used trig ratios to solve word problems (Kite Example)
Solving Trig Equations
We solved for simple trig equations by using our special acute angles.
We solved for trig equations by using our trig inverse function and finding
𝜃𝑅 𝑜𝑟 𝑥𝑅
We also encountered cases where we had to rearrange our equation and
isolate our trig function.
We used factoring or the quadratic formula to solve trig equations.
When we solved Trig Equations we must note the restriction on our
solution:
- 0 ≤ 𝜃 ≤ 2𝜋
- No restrictions (infinite solutions)
- - other restrictions (we added or subtracted 2𝜋)
Graphing Trig Functions
We sketched the graphs of all 6 trig functions
We found the characteristics of each of these trig functions:
- Max/Min
- Amplitude
- Period
- Intercepts
- Asymptotes
Transformations of Sinusoidal
Functions
The general form of a Sinusoidal Function can be written as:
𝑦 = a • sin 𝑘 𝑥 − 𝑐 + 𝑑
OR
𝑦 = a • cos 𝑘 𝑥 − 𝑐 + 𝑑
We discussed the effects of variables a, k, c and d on the function
We used mapping notation to graph our transformed functions
-
(x, 𝑦)
1
𝑘
( 𝑥 + 𝑐, 𝑎𝑦 + 𝑑)
We found the 5 key points for both sine and cosine functions
We also used formulas of amplitude, period, vertical shift as well as our
knowledge of the behaviour of sine and cosine functions to find the
equation in general form
Applications of Sinusoidal
Functions
We applied sine and cosine functions to real life scenarios where we:
• Sketched the function
• Found the characteristics of the function
• Found the equation of the function that would model the scenario
• Solved for specific values of the function