2. 𝑦 = 𝐴 sin 𝐵(𝑥 + 𝐶) + 𝐷
𝑦 = 𝐴 cos 𝐵(𝑥 + 𝐶) + 𝐷
We already know:
𝐴 is the Amplitude
𝐵 is the Frequency
𝐷 is the Vertical Shift
𝐶 is the Horizontal Shift. However,
+ 𝐶 is shift left (e.g. timing started later)
− 𝐶 is shift right (e.g. object started moving later)
3. 𝑦 = 𝐴 sin 𝐵(𝑥 + 𝐶) + 𝐷
𝑦 = 𝐴 cos 𝐵(𝑥 + 𝐶) + 𝐷
We already know:
𝐴 is the Amplitude
𝐵 is the Frequency
𝐷 is the Vertical Shift
𝐶 is the Horizontal Shift. However,
+ 𝐶 is shift left (e.g. timing started later)
− 𝐶 is shift right (e.g. object started moving later)
4. 𝐶 is the Horizontal Shift
+ 𝐶 is shift left
− 𝐶 is shift right
Once you have found C and have your complete equation, you will
need to find all the points where the graph reaches a particular
height.
E.g. try typing 𝑦 = 2 cos 3 𝑥 − 2 + 1 into desmos and find as
many 𝑥 values as you can where 𝑦 = 2 (see image above)
Notice that if you scroll across, there is an infinite amount of answers to
write. It is impossible to write them all individually
So, we need a general solution. This gives all possible answers
written as one answer.
5. The General Solution - Cosine Overview
We need to set 𝑦 to the given value and then rearrange the equation to
make 𝑥 the subject.
For example, to find the general solution of:
𝑦 = 2 cos 3 𝑥 − 2 + 1 where 𝑦 = 2
We start with:
2 cos 3 𝑥 − 2 + 1 = 2
and rearrange it to:
𝑥 =
2𝑛𝜋
3
± 0.3491 + 2
The 𝑛 represents any integer, so we can find each and every one of the
infinite solutions by substituting in different values for 𝑛
Formula Sheet
6. The General Solution - Cosine Overview
Step-by-step solving of the previous slide example
1. 2 cos 3 x − 2 + 1 = 2
2. 2 cos 3 x − 2 = 1
3. cos 3 x − 2 =
1
2
4. 3 x − 2 = cos−1 1
2
5. 3 𝑥 − 2 = 1.0472
6. 3 𝑥 − 2 = 2𝑛𝜋 ± 1.0472
7. 𝑥 − 2 =
2𝑛𝜋
3
± 0.3491
8. 𝑥 =
2𝑛𝜋
3
± 0.3491 + 2
÷ 3
÷ 3
−1
÷ 2
cos−1
1
2
𝛼
+2
7. The General Solution - Sine Overview
We need to set 𝑦 to the given value and then rearrange the equation to
make 𝑥 the subject.
For example, to find the general solution of:
𝑦 = 2 sin 3 𝑥 − 2 + 1 where 𝑦 = 2
We start with:
2 sin 3 𝑥 − 2 + 1 = 2
and rearrange it to:
𝑥 =
𝑛𝜋
3
+ −1 𝑛(0.1745) + 2
The 𝑛 represents any integer, so we can find each and every one of the
infinite solutions by substituting in different values for 𝑛
9. The General Solution – Step-by-step
The step numbers match the step-by-step working on slides 5 and 7
1. Remove the 𝑦 = and instead write … = 𝑦−𝑣𝑎𝑙𝑢𝑒 𝑔𝑖𝑣𝑒𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑞𝑢𝑒𝑠𝑡𝑖𝑜𝑛
2. Subtract D from both sides
3. Divide both sides by A
4. Take cos−1
or sin−1
of both sides (cos−1
cancels out cos, and sin−1
cancels
out sin )
5. You now have 𝜃 = 𝛼 (the left side is 𝜃 and the right side is 𝛼)
6. Check the formula sheet for the general solution formula:
if you had sin then the general solution is: 𝜃 = 𝑛𝜋 + −1 𝑛𝛼
if you had cos then the general solution is: 𝜃 = 2𝑛𝜋 ± 𝛼
7. Divide both sides by B (carefully divide all terms on the right by B)
8. Subtract/Add C to both sides (leave the subtraction/addition on
the end of the equation)
You should now have 𝑥 = 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 𝑛, and you are finished!
10. Practice: Find the General Solution
1. 𝑦 = 4 cos 2 𝑥 + 3 − 5 where 𝑦 = −3
2. 𝑦 = 4 sin 2 𝑥 + 3 − 5 where 𝑦 = −3
3. 𝑦 =
1
2
cos
𝜋
2
𝑥 − 1 + 2 where 𝑦 =
5
2
4. 𝑦 =
1
2
sin
𝜋
2
𝑥 − 1 + 2 where 𝑦 =
5
2
5. 𝑦 = 5 cos
2𝜋
3
𝑥 + 4 + 7 where 𝑦 = 10
6. 𝑦 = 5 sin
2𝜋
3
𝑥 + 4 + 7 where 𝑦 = 10
