Sin 22.5° Cos 22.5° (Trig Identity Guide)
The exact values are (\sin 22.5^\circ=\frac{\sqrt{2-\sqrt2}}{2}) and (\cos 22.5^\circ=\frac{\sqrt{2+\sqrt2}}{2}). To find them, treat (22.5^\circ) as half of (45^\circ), use the half-angle identities, and choose positive roots because (22.5^\circ) lies in Quadrant I. The product is (\sin22.5^\circ\cos22.5^\circ=\frac{\sqrt2}{4}).
If you are searching for these values, you may have found a formula full of plus and minus signs without a clear guide to choosing between them. The key is to connect the angle to a familiar one: (22.5^\circ) is half of (45^\circ). From there, the half-angle identities give exact answers, not just decimal estimates. I’ll show each step and a few checks to help you spot common mistakes.
Angle Setup and Half-Angle Identity Overview
A half-angle identity finds the sine or cosine of half an angle using the cosine of the original angle. Since (22.5^\circ=45^\circ/2), the needed starting value is (\cos45^\circ=\frac{\sqrt2}{2}). The formulas include a sign choice, so we also need to check the quadrant.
Why is (22.5^\circ) a half-angle?
A half-angle is an angle made by dividing a known angle by two. Here, (45^\circ\div2=22.5^\circ), so we can use the half-angle formulas with (\theta=45^\circ). This is useful because sine and cosine at (45^\circ) are known exactly.
In these identities, (\theta) is the original, larger angle, and (\theta/2) is the angle whose sine or cosine we want. Put (45^\circ) in place of (\theta), and the formulas will give values for (22.5^\circ).
What are the half-angle identities?
The half-angle identities express the sine and cosine of (\theta/2) in terms of (\cos\theta). The square root has a plus-or-minus sign because the sign depends on the quadrant of the half-angle. For (22.5^\circ), both results are positive.
[ \sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos\theta}{2}} ]
[ \cos\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1+\cos\theta}{2}} ]
The minus sign appears in the sine formula’s numerator; the plus sign appears in the cosine formula’s numerator. Those signs are part of the identities, not instructions to always use a negative or positive answer. You select the sign outside the square root by checking where (\theta/2) lies.
How do we choose the signs?
The quadrant is the part of the coordinate plane that contains an angle in standard position. An angle of (22.5^\circ) is between (0^\circ) and (90^\circ), so it lies in Quadrant I. In this quadrant, both sine and cosine are positive, which tells us to use the positive square roots.
A quick sign check is worth doing before simplifying. If you skip it, the algebra can be correct while the final value has the wrong sign. For this angle, the positive choices match the fact that both coordinates on the unit circle are positive.
Derivation, Simplification, and Exact-Value Requirements
To derive the exact values, substitute (\theta=45^\circ) into each identity and use (\cos45^\circ=\frac{\sqrt2}{2}). Then simplify the expressions under the square roots. Keeping fractions clear at each step helps prevent errors with the factor of (2).
Deriving (\sin22.5^\circ)
The sine half-angle identity uses (1-\cos\theta). With (\theta=45^\circ), and with the positive sign selected from the quadrant, the substitution is:
[ \sin22.5^\circ =\sqrt{\frac{1-\cos45^\circ}{2}} =\sqrt{\frac{1-\frac{\sqrt2}{2}}{2}}. ]
To simplify, rewrite (1) as (\frac{2}{2}). Then divide the resulting fraction by (2):
[ \sqrt{\frac{1-\frac{\sqrt2}{2}}{2}} =\sqrt{\frac{\frac{2-\sqrt2}{2}}{2}} =\sqrt{\frac{2-\sqrt2}{4}} =\frac{\sqrt{2-\sqrt2}}{2}. ]
Therefore,
[ \boxed{\sin22.5^\circ=\frac{\sqrt{2-\sqrt2}}{2}}. ]
The radical is exact: it gives the value without rounding. The expression (2-\sqrt2) is positive, so its square root is real.
Deriving (\cos22.5^\circ)
For cosine, the half-angle identity uses (1+\cos\theta). Substitute (\cos45^\circ=\frac{\sqrt2}{2}), and again choose the positive root because the angle is in Quadrant I:
[ \cos22.5^\circ =\sqrt{\frac{1+\cos45^\circ}{2}} =\sqrt{\frac{1+\frac{\sqrt2}{2}}{2}}. ]
Rewrite (1) as (\frac{2}{2}), then simplify:
[ \sqrt{\frac{1+\frac{\sqrt2}{2}}{2}} =\sqrt{\frac{\frac{2+\sqrt2}{2}}{2}} =\sqrt{\frac{2+\sqrt2}{4}} =\frac{\sqrt{2+\sqrt2}}{2}. ]
So the exact cosine is
[ \boxed{\cos22.5^\circ=\frac{\sqrt{2+\sqrt2}}{2}}. ]
The two answers look similar, but do not swap the inner signs. Sine comes from (1-\cos\theta); cosine comes from (1+\cos\theta).
Exact values and decimal approximations
An exact value is a form, often involving fractions and radicals, that represents a number without rounding. A decimal approximation is useful for checking or estimating, but it usually does not replace the exact form when a problem asks for an exact answer.
| Quantity | Exact value | Decimal check |
|---|---|---|
| (\sin22.5^\circ) | (\frac{\sqrt{2-\sqrt2}}{2}) | (\approx0.382683) |
| (\cos22.5^\circ) | (\frac{\sqrt{2+\sqrt2}}{2}) | (\approx0.923880) |
| (\sin22.5^\circ\cos22.5^\circ) | (\frac{\sqrt2}{4}) | (\approx0.353553) |
The decimals are rounded to six places. A calculator may show slightly different final digits if it rounds differently, but the exact radical forms do not change.
Verification Checks and Common Sign Errors
A verification check tests whether an answer fits a known identity or a reasonable estimate. For these values, use the Pythagorean identity, a double-angle identity, and approximate size. Each check catches a different kind of mistake, especially an incorrect sign or a missing factor of (2).
Check the Pythagorean identity
For every angle (x), the identity (\sin^2x+\cos^2x=1) must hold. Squaring the exact values gives:
[ \sin^2 22.5^\circ=\frac{2-\sqrt2}{4}, \qquad \cos^2 22.5^\circ=\frac{2+\sqrt2}{4}. ]
Add them:
[ \frac{2-\sqrt2}{4}+\frac{2+\sqrt2}{4} =\frac{4}{4}=1. ]
The (\sqrt2) terms cancel, so the values pass the check. Notice that this check confirms the squared values; the quadrant check is still needed to confirm the signs.
Check the product with a double-angle identity
The double-angle identity says (\sin(2x)=2\sin x\cos x). Set (x=22.5^\circ), so (2x=45^\circ):
[ \sin45^\circ=2\sin22.5^\circ\cos22.5^\circ. ]
Since (\sin45^\circ=\frac{\sqrt2}{2}), divide by (2):
[ \sin22.5^\circ\cos22.5^\circ=\frac{\sqrt2}{4}. ]
This verifies the product directly. It is also a useful shortcut when a question asks only for the product, not for each value separately.
Common mistakes to avoid
The half-angle steps are short, but a few slips can change the answer. Check the identity, the sign, and the simplification before writing a final result.
| Mistake | Why it causes trouble | Better check |
|---|---|---|
| Choosing a sign without checking the quadrant | The half-angle formulas allow either sign, depending on the angle. | Confirm that (22.5^\circ) is in Quadrant I. |
| Using (1+\cos45^\circ) for sine | The sine formula uses (1-\cos\theta). | Write both identities before substituting. |
| Forgetting to divide by (2) | The denominator (2) is part of each formula. | Keep parentheses around the full fraction. |
| Stopping at a decimal | A rounded value is not an exact radical form. | Give the simplified radical, then use a decimal only to check. |
| Losing the outer denominator | (\sqrt{A/4}=\sqrt A/2). | Take the square root of both numerator and denominator. |
A quick estimate helps too. Sine is about (0.383), while cosine is about (0.924), so cosine should be larger. If your results reverse those sizes, review the substitutions.
Applications in Exact Trigonometric Evaluation
Exact values are useful when a problem asks for a result without rounding or when later steps involve symbolic algebra. The values for (22.5^\circ) can be used alone, multiplied together, or checked against related identities. The key is to match the method to what the question asks.
When should you use the half-angle identities?
Use the half-angle identities when the angle you need is half of an angle with a known cosine. For (22.5^\circ), the known angle is (45^\circ). This route is especially direct when the problem asks for exact sine or cosine values.
If you need the product (\sin22.5^\circ\cos22.5^\circ), the double-angle identity is shorter. If you need both sine and cosine separately, use the half-angle identities, then check the product if desired.
| What the question asks | Efficient method | Exact result |
|---|---|---|
| Find (\sin22.5^\circ) | Sine half-angle identity | (\frac{\sqrt{2-\sqrt2}}{2}) |
| Find (\cos22.5^\circ) | Cosine half-angle identity | (\frac{\sqrt{2+\sqrt2}}{2}) |
| Find their product | Double-angle identity | (\frac{\sqrt2}{4}) |
| Check both values | Pythagorean identity | Their squares add to (1) |
How to present a complete solution
A complete solution shows enough work for someone else to follow the reasoning. State the angle relationship, write the identity, substitute the known value, choose the sign using the quadrant, and simplify. This also makes it easier to find a mistake if your answer differs from a key.
For example, a concise sine solution could read: “Since (22.5^\circ=45^\circ/2) and (22.5^\circ) is in Quadrant I, use the positive sine half-angle root. Substituting (\cos45^\circ=\frac{\sqrt2}{2}) gives (\sin22.5^\circ=\sqrt{\frac{1-\sqrt2/2}{2}}=\frac{\sqrt{2-\sqrt2}}{2}).” The same pattern works for cosine, with a plus sign inside the formula.
Conclusion
The exact sine and cosine of (22.5^\circ) come from halving (45^\circ), using (\cos45^\circ=\frac{\sqrt2}{2}), and choosing positive roots because the angle is in Quadrant I. The results are (\frac{\sqrt{2-\sqrt2}}{2}) for sine and (\frac{\sqrt{2+\sqrt2}}{2}) for cosine. Check them with (\sin^2x+\cos^2x=1).
Frequently Asked Questions
These answers recap the key values, formulas, and checks for (22.5^\circ). They are useful when you need a quick result, but the quadrant check still matters whenever you apply a half-angle identity to a different angle.
What is the exact value of (\sin22.5^\circ)?
The exact value is (\frac{\sqrt{2-\sqrt2}}{2}). It follows from the sine half-angle identity with (\theta=45^\circ), using (\cos45^\circ=\frac{\sqrt2}{2}). The positive root is correct because (22.5^\circ) lies in Quadrant I.
What is the exact value of (\cos22.5^\circ)?
The exact value is (\frac{\sqrt{2+\sqrt2}}{2}). Use the cosine half-angle identity with (\theta=45^\circ), then substitute (\cos45^\circ=\frac{\sqrt2}{2}). Since (22.5^\circ) is in Quadrant I, cosine is positive.
What is (\sin22.5^\circ\cos22.5^\circ)?
The product is (\frac{\sqrt2}{4}). Apply (\sin(2x)=2\sin x\cos x) with (x=22.5^\circ). Then (\sin45^\circ=2\sin22.5^\circ\cos22.5^\circ), and dividing (\frac{\sqrt2}{2}) by (2) gives the result.
Why are both values positive?
Both values are positive because (22.5^\circ) is between (0^\circ) and (90^\circ), which places it in Quadrant I. In that quadrant, sine and cosine are both positive. The quadrant determines which sign to use in the half-angle formulas.
Why does the half-angle formula have a plus-or-minus sign?
The identity gives a square root, and a square root can correspond to a positive or negative value before the angle’s location is considered. The correct sign depends on the quadrant of the half-angle. For (22.5^\circ), choose positive for both sine and cosine.
Can I use decimals instead of exact radical values?
Use decimals when an estimate is enough or when checking your work. The approximate values are (\sin22.5^\circ\approx0.382683) and (\cos22.5^\circ\approx0.923880). If a problem asks for an exact value, give the radical form rather than only a rounded decimal.
How can I check the two exact values?
Square each value and add them. Their squares are (\frac{2-\sqrt2}{4}) and (\frac{2+\sqrt2}{4}); the sum is (1). This matches (\sin^2x+\cos^2x=1). Also confirm the signs from the quadrant, since squaring removes sign information.
Which identity should I use if only the product is requested?
Use the double-angle identity, not two separate half-angle calculations. Since (2(22.5^\circ)=45^\circ), the equation (\sin45^\circ=2\sin22.5^\circ\cos22.5^\circ) gives the product as (\frac{\sqrt2}{4}).
(This article was written by one of our staff writers, Alan Westbrook. Visit our Meet the Team page.)
