Given cos x = 3/4 with x acute, sin 2x is 0.9922, exactly 3√7/8. The angle x is 41.41° and you never work it out. Double it instead and you get 1.3229, which is not a sine at all.
The ceiling at 1 is not a convention. A sine is the height of a point on a circle of radius 1, so nothing above the line can be one.
A point on the unit circle is (cos x, sin x) and it is 1 away from the origin. Pythagoras on that right-angled triangle gives, at once:
cos²x + sin²x = 1
That is the Pythagorean identity, and it really is just Pythagoras with the legs named. It holds for every x, which is what makes it an identity rather than an equation: there is nothing to solve, and you may substitute it in either direction whenever you like.
The form you will use most is the rearranged one, because questions give you one ratio and want another:
sin x = ±√(1 − cos²x)
The ± is the whole difficulty, and the question always resolves it. "x is acute" means the first quadrant, so take the positive root. "x is obtuse" means the second, where the sine is still positive but the cosine is not. Never discard the sign silently; write down which quadrant you are in and why, because that is frequently a mark.
There are three to carry, and the cosine one has three faces:
| Identity | Use it when |
|---|---|
| sin 2x = 2 sin x cos x | You have both ratios, or can get both. |
| cos 2x = 2cos²x − 1 | You have the cosine only. The most useful form. |
| cos 2x = 1 − 2sin²x | You have the sine only. |
| cos 2x = cos²x − sin²x | You have both and want symmetry. |
The three cosine forms are the same statement with the identity substituted in, so they always agree. For cos x = 3/4 all three give 0.125, exactly 1/8. Pick whichever form matches what the question handed you and you save a line of work.
"Double angle" does not mean "double the answer". sin 2x is the sine of twice the angle, and that is not twice the sine. With cos x = 3/4 and x acute:
The last of those is not a near miss. It is bigger than 1, so it is not the sine of anything. That is worth more than remembering the identity, because it is a check you can run on your own answer in two seconds: if a sine or cosine comes out above 1 or below −1, the method is wrong, not the arithmetic.
Move the slider and watch when the mistake becomes impossible rather than merely wrong. 2 sin x passes the ceiling as soon as sin x passes 0.5, which is at cos x = √3/2 = 0.866. Below that, every version of this error is detectable without knowing the right answer.
Given cos x = 3/4 and that x is acute, find sin 2x and cos 2x.
x never appeared. It is 41.41° if you want it, and finding it would have cost an inverse cosine, a rounding, and then two more trigonometric evaluations on a rounded number. The identities exist so that you can answer a question about 2x from information about x, exactly, without the angle in between.
That is also why Paper 1 can ask this. There is no calculator, 41.41° is unobtainable, and the answer is a surd that comes out of exact fractions. If you find yourself wanting an inverse cosine on Paper 1, the identity you need is the one you have not used.
tan x comes free. Once you have both ratios, tan x = sin x / cos x = (√7/4) ÷ (3/4) = √7/3 = 0.8819. The 4s cancel, which they always will, so a tangent from a cosine is two steps: Pythagoras for the sine, then divide. Dividing the surds before converting to a decimal is what keeps it exact.
The machine cannot do this question. It can only check your answer, and that is genuinely worth doing, because an identity applied with one sign wrong gives a plausible number rather than an error.
When you may use it. Analysis Paper 1 is non-calculator, and identity questions live there, because the answer is a surd. On Paper 2 use the machine to check, not to solve: a question that gives you cos x = 3/4 wants the identity, and a decimal from an inverse cosine will usually fail to match the exact answer the mark scheme wants.
The mark people lose. Taking the inverse cosine, rounding it, and working from the rounded angle. It usually gets within a rounding of the right answer and sometimes does not, and on Paper 1 it is not available at all. It also throws away the exact form the question was built to produce. The habit: if the question gives you a ratio as a fraction, the answer is a surd and the angle is not part of the method.
Throughout: cos x = 3/4 and x is acute.
1. Find sin x, to 4 decimal places.
2. Find sin 2x, to 4 decimal places.
3. Find cos 2x, to 3 decimal places.
4. Find tan x, to 4 decimal places.
5. A student answers sin 2x = 1.3229. Without working out the right answer, how do you know that is wrong?
1 markUsing the Pythagorean identity to get the second ratio.
1 markChoosing the sign from the quadrant, and saying so.
1 markThe correct double angle identity, stated before it is used.
1 markThe answer in exact form on Paper 1.
Writing the identity down before substituting into it is worth doing even when you are confident. It earns the method mark on its own, so a slip in the arithmetic afterwards costs one mark instead of two.
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