Key takeaway: This precalculus trig identities lesson explains which equations hold for every angle and how to use them. An identity is true wherever both sides are defined, and the core one is \(\sin^2\theta+\cos^2\theta=1\). The page also covers sum, difference and double-angle formulas under standard F-TF.C.8. In addition, it shows how to verify an identity by changing only one side.
A trigonometric identity is an equation that holds for every angle where both sides are defined. The key identities are the reciprocal and quotient identities, the Pythagorean identity sin²θ + cos²θ = 1, the sum and difference formulas for sine, cosine and tangent, and the double-angle formulas that follow from them.
Precalculus Trig Identities Lesson: key facts
- Standards F-TF.C.8 and F-TF.C.9: prove and use the Pythagorean identity, and the addition formulas for sine, cosine and tangent.
- Pythagorean identities: \(\sin^{2}\theta+\cos^{2}\theta=1\), \(1+\tan^{2}\theta=\text{sec}^{2}\theta\) and \(1+\text{cot}^{2}\theta=\text{csc}^{2}\theta\).
- Sum formulas: \(\sin(A+B)=\sin A\cos B+\cos A\sin B\) and \(\cos(A+B)=\cos A\cos B-\sin A\sin B\).
- Double angle: \(\sin 2\theta=2\sin\theta\cos\theta\) and \(\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta\).
- To verify an identity, work on one side only, rewrite in sine and cosine, and simplify until it matches the other side.
What are the basic trigonometric identities?
An identity is true for every angle where both sides are defined. The point at angle \(\theta\) on the unit circle is \((\cos\theta,\sin\theta)\) and lies 1 unit from the origin, so \(\cos^{2}\theta+\sin^{2}\theta=1\).
Dividing both sides of the Pythagorean identity by \(\cos^{2}\theta\) or by \(\sin^{2}\theta\) produces the other two Pythagorean identities. Memorize one and derive the rest when you need them. The sign of each function depends on the quadrant, so always check which quadrant the angle is in before choosing a square root.
| Type | Identities |
|---|---|
| Reciprocal | \(\text{csc}\,\theta=\frac{1}{\sin\theta}\), \(\text{sec}\,\theta=\frac{1}{\cos\theta}\), \(\text{cot}\,\theta=\frac{1}{\tan\theta}\) |
| Quotient | \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), \(\text{cot}\,\theta=\frac{\cos\theta}{\sin\theta}\) |
| Pythagorean | \(\sin^{2}\theta+\cos^{2}\theta=1\), \(1+\tan^{2}\theta=\text{sec}^{2}\theta\), \(1+\text{cot}^{2}\theta=\text{csc}^{2}\theta\) |
How do you verify an identity?
- Pick the more complicated side and work only on that side.
- Rewrite every function in terms of sine and cosine.
- Combine fractions, factor, or substitute a Pythagorean identity.
- Stop when the side matches the other one.
Never move terms across the equal sign, because that assumes the identity is true. If both sides look complicated, simplify each side separately until they reach the same expression.
Example. Verify \(\tan\theta+\frac{\cos\theta}{1+\sin\theta}=\text{sec}\,\theta\). Rewrite the left side as \(\frac{\sin\theta(1+\sin\theta)+\cos^{2}\theta}{\cos\theta(1+\sin\theta)}\). The numerator is \(\sin\theta+\sin^{2}\theta+\cos^{2}\theta=\sin\theta+1\), so the fraction reduces to \(\frac{1}{\cos\theta}=\text{sec}\,\theta\).
What are the sum and difference formulas?
| Function | Sum | Difference |
|---|---|---|
| Sine | \(\sin A\cos B+\cos A\sin B\) | \(\sin A\cos B-\cos A\sin B\) |
| Cosine | \(\cos A\cos B-\sin A\sin B\) | \(\cos A\cos B+\sin A\sin B\) |
| Tangent | \(\frac{\tan A+\tan B}{1-\tan A\tan B}\) | \(\frac{\tan A-\tan B}{1+\tan A\tan B}\) |
Because \(75^{\circ}=45^{\circ}+30^{\circ}\), \(\sin 75^{\circ}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx 0.966\). Note that \(\sin(A+B)\neq\sin A+\sin B\).
The cosine formulas swap the sign between the two products, and the tangent formulas divide by \(1-\tan A\tan B\) in the sum case. Special angles such as \(15^{\circ}\), \(75^{\circ}\) and \(105^{\circ}\) get exact values by splitting them into \(30^{\circ}\), \(45^{\circ}\) and \(60^{\circ}\).
What are the double-angle formulas?
Setting \(B=A\) in the sum formulas gives \(\sin 2\theta=2\sin\theta\cos\theta\) and \(\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta\), and \(\tan 2\theta=\frac{2\tan\theta}{1-\tan^{2}\theta}\). If \(\sin\theta=\frac{3}{5}\) in Quadrant I, then \(\cos\theta=\frac{4}{5}\), \(\sin 2\theta=\frac{24}{25}\) and \(\cos 2\theta=\frac{7}{25}\).
The three cosine forms are interchangeable. Choose \(2\cos^{2}\theta-1\) when only the cosine is known and \(1-2\sin^{2}\theta\) when only the sine is known. The graph shows that \(2\sin x\cos x\) and \(\sin 2x\) coincide.
Trigonometric Identities: frequently asked questions
What is the difference between an identity and an equation?
An identity is true for every angle where both sides are defined, such as \(\sin^{2}\theta+\cos^{2}\theta=1\). An ordinary equation like \(\sin\theta=\frac{1}{2}\) is true only for certain angles.
Why can I not move terms across the equal sign when verifying an identity?
Moving terms assumes the identity is already true, which is the very thing to be shown. Work on one side until it becomes the other, or simplify both sides separately to the same expression.
Is sin(A + B) equal to sin A + sin B?
No. For \(A=B=60^{\circ}\), \(\sin 120^{\circ}\approx 0.866\) but \(\sin 60^{\circ}+\sin 60^{\circ}\approx 1.732\). The correct formula is \(\sin A\cos B+\cos A\sin B\).
Which form of the cosine double-angle formula should I use?
Use \(2\cos^{2}\theta-1\) when you know the cosine, \(1-2\sin^{2}\theta\) when you know the sine, and \(\cos^{2}\theta-\sin^{2}\theta\) when you know both.
How do I choose the sign when I take a square root in an identity problem?
Use the quadrant of the angle. Sine is positive in Quadrants I and II, cosine is positive in Quadrants I and IV, and tangent is positive in Quadrants I and III.
How do you find a missing trig value from one identity?
Suppose \(\sin\theta=\frac{3}{5}\) with \(\theta\) in Quadrant I. Then \(\cos^2\theta=1-\frac{9}{25}=\frac{16}{25}\), so \(\cos\theta=\frac{4}{5}\). In Quadrant II, however, the same square gives \(-\frac{4}{5}\). Therefore the quadrant picks the sign, and the identity picks the size.
How do you verify an identity step by step?
Start with the more complicated side. For \(\frac{\sin\theta}{1-\cos^2\theta}\), replace the denominator with \(\sin^2\theta\). Next, cancel one factor to get \(\frac{1}{\sin\theta}=\csc\theta\). Never move terms across the equal sign, because that assumes the claim you want to prove.
What do sum and difference formulas give in a precalculus trig identities lesson?
Write 75 degrees as \(45^{\circ}+30^{\circ}\). Then \(\sin 75^{\circ}=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}\). A common slip is to claim \(\sin(A+B)=\sin A+\sin B\), and a quick test with \(A=B=90^{\circ}\) shows why that fails.
Which double-angle form should you choose?
Use \(\cos2\theta=2\cos^2\theta-1\) when you know cosine. Use \(1-2\sin^2\theta\) when you know sine. For \(\sin\theta=\frac{3}{5}\), the second form gives \(\frac{7}{25}\) quickly.
What should you practice after learning these formulas?
Verify three identities and write each step on a separate line. Then compute \(\sin 15^{\circ}\) and \(\cos 105^{\circ}\) from special angles. Finally, test yourself with the quiz, and save the longer proofs for the worksheet.
See every chapter on the Precalculus math page.
