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The fundamental identity and angle-addition formulas connect sine and cosine at different angles and enable equivalent transformations.
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Formula
Angles in radians or degrees; sine and cosine are dimensionless.
| Symbol | Meaning | Unit |
|---|---|---|
| Angles that must use the same angular unit | ° or rad | |
| sin, cos | Sine and cosine functions | dimensionless |
\sin^2x+\cos^2x=1,\quad \sin(a\pm b)=\sin a\cos b\pm\cos a\sin b,\quad \cos(a\pm b)=\cos a\cos b\mp\sin a\sin bThe identities follow from the unit circle and the angle-addition formulas. Read the paired signs carefully: in sin(a±b), the second term keeps the sign; in cos(a±b), it takes the opposite sign. The fundamental identity holds for every real x. The identities are exact, but the angles must be supplied consistently in one unit.
The fundamental identity holds for every real angle. Angle-addition formulas require both angles to use the same unit.
Coordinates on the unit circle and rotation by an angle yield the identities.
Cosine of a sum
For cosine, the sign between product terms is opposite to the sign inside the angle.
Sine from cosine
Follows directly from sin2x+cos2x=1.
Find sin(75°) using the angle-addition identity.
sin(30°+45°)=sin30°cos45°+cos30°sin45°=(√2+√6)/4≈0.966.
Using the same sign in the cosine formula as in the angle parentheses.
cos(a±b)=cos a cos b∓sin a sin b: the sine-product term has the opposite sign.
Formula cluster
Identities connect the unit circle with angle and triangle relations.
For a=30° and b=45°, sin(75°)=sin30°cos45°+cos30°sin45°=(√2+√6)/4≈0.966. Check: this agrees with sin75°, and sin275°+cos275°=1.
Simplify angle expressions, solve trigonometric equations, and transform oscillation or wave relations.
Here is how to work through a typical Trigonometric identities (sin²x + cos²x = 1; Additionstheoreme) task step by step:
Task
Find sin(75°) using the angle-addition identity.
Solution path
sin(30°+45°)=sin30°cos45°+cos30°sin45°=(√2+√6)/4≈0.966.
They let you rewrite the same expression in a form better suited to a problem. Angle-addition formulas split an angle such as 75° into familiar angles, while sin2x+cos2x=1 replaces one function with the other. This helps with exact values, equations, and oscillation or wave models. Check signs and angle units before substitution. The unit-circle definition and angle-addition formulas are exact relations; rounded table values are only a later approximation. In equations, a suitable form also makes it easier to isolate the unknown angle. Equivalent transformations remain exact.
For cosine, the sin a sin b product carries the opposite sign: cos(a+b)=cos a cos b−sin a sin b. Check with a=b=45°: cos90°=0, and the minus form gives ½−½=0; a plus would be wrong. For cos(a−b), the sign correspondingly becomes plus. Check signs and angle units before substitution. The unit-circle definition and angle-addition formulas are exact relations; rounded table values are only a later approximation. The pattern stays consistent; if unsure, check it with simple angles such as 0°. Symmetry checks also help.
Yes. On the unit circle, the point at angle x has coordinates (cos x,sin x). Its distance from the origin is 1, so Pythagoras gives cos2x+sin2x=1. The identity therefore also holds for negative angles, large angles, and angles beyond one full turn; sine and cosine are periodic. Check signs and angle units before substitution. The unit-circle definition and angle-addition formulas are exact relations; rounded table values are only a later approximation. Rearranging also gives sin2x=1−cos2x or cos2x=1−sin2x. This lets you eliminate one function.
No. Trigonometric values depend on the stated angle unit. Before using an addition identity, both angles must use the same unit. In calculus formulas such as the derivative of sin x, x is conventionally in radians. Convert degrees with xrad=xdeg·π/180 or set the calculator to the correct mode. Check signs and angle units before substitution. The unit-circle definition and angle-addition formulas are exact relations; rounded table values are only a later approximation. Radians let calculus derivatives and limit formulas use their standard form without extra conversion factors.
Write 75° as 30°+45°. Then sin75°=sin30°cos45°+cos30°sin45°. Using sin30°=1/2, cos30°=√3/2 and sin45°=cos45°=√2/2 gives (√2+√6)/4≈0.966. The exact radical is more precise than the rounded decimal and can be checked by comparing its square with 1−cos275°. Check signs and angle units before substitution. The unit-circle definition and angle-addition formulas are exact relations; rounded table values are only a later approximation. Multiplying the radicals gives an exact value: (√6+√2)/4 is not a rounded decimal and preserves the full information. Use the decimal only as a check.
Quanta exam set
Curated exam set for "Trigonometric identities":
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Which formula describes Trigonometric identities?
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Question (front)
How do you rearrange Trigonometric identities for Cosine of a sum?
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Question (front)
Which common mistake happens with Trigonometric identities?
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