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Core 2 Revision
Trigonometrical identities and simple equations
The two identities that you need to know:
Sin2 x + cos2 x = 1
Tan x = sin x / cos x
These are key to answering any kind of question as they usually involve simplifying
expressions so that they only involve one of the three functions (sin, cos and tan).
2cos2 x + 9sin x = 3sin2 x
2(1 sin2 x) + 9sin x = 3sin2 x
2 2sin2 x + 9sin x = 3sin2x
5sin2 x 9sin x 2 = 0
Solving for x
Solving equations can be quite straight forward, as long as you follow the following steps.
1. Rearrange the equation so that there is only one function involved.
2. Work out your value of x
For example: sin x = ½ so x = 30
3. Draw a Cast Diagram and use it draw the value of x in each of the four quadrants.
4. If the value of sin x is positive consider the quadrants where sin is positive (the same
goes for cos and tan, if cos x is positive only consider the quadrants where cos is
If the value of sin x was negative consider the quadrants where sin is negative
5. Work out the values of x for where the function is positive.
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Rules for using the Cast Diagram
1. Always start at the point labelled `0' above when working out the solution.
2. If the angle is positive (e.g. 30) then go anticlockwise from the start point.
3. If the angle is negative (e.g. -30) then go clockwise from the start point.…read more
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Sin x = ½
X = 30 (inverse sin it)
Draw out your cast diagram (look at the diagram above, sin is positive in the first and second
This cast diagram shows all of the possible solutions, however because we know that the
value of the function is positive we only look where sin is positive, which is the first and third