Key takeaway: This precalculus trig identities quiz poses ten multiple-choice items about squared sine and cosine, angle sums, angle differences and doubled angles. Exactly one choice is correct, and each explanation names the usual mistake. Standard F-TF.C.8 shapes the items. Three questions include a figure, so you can practice reading a unit-circle point. The quiz takes about 15 minutes, and it works as a warm-up before the longer test.
Ten multiple-choice questions on trigonometric identities: Pythagorean and reciprocal identities, sum and difference formulas, and double angles. Choose the single best answer, then read the explanation to learn the typical mistake.
🎉 Quiz complete!
Remember:
- 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 do questions 1 to 4 check?
Question 1 gives \(\sin\theta=\frac{12}{13}\) for a Quadrant II point with \(x=-\frac{5}{13}\). Next, \(\sin\theta\cot\theta\) simplifies to \(\cos\theta\) in question 2. Question 3 asks which equation is a true identity, and \(1+\tan^2\theta=\sec^2\theta\) is the one. Finally, \(\sin\theta=\frac{3}{5}\) leads to \(\cos2\theta=\frac{7}{25}\) in question 4.
How do you evaluate questions 5 and 6?
For \(\sin75^{\circ}\), the rays at 45 and 75 degrees hint at \(45^{\circ}+30^{\circ}\). Then \(\tan(A+B)\) with \(\tan A=\frac{1}{2}\) and \(\tan B=\frac{1}{3}\) equals \(\frac{5/6}{5/6}=1\).
How can you retake the precalculus trig identities quiz?
Solve before you view the options. After that, test each wrong option with a value such as \(\theta=30^{\circ}\). For example, \(\frac{\sin\theta}{1-\cos^2\theta}\) in question 7 equals \(\csc\theta\), while the other choices fail at one angle.
What do questions 8 to 10 add?
The graph in question 8 has period \(\pi\), so \(2\sin x\cos x=\sin2x\). Question 9 gives \(\tan\theta=\frac{3}{4}\) in Quadrant III, and question 10 asks for \(\sin A\cos B-\cos A\sin B\).
What is the next move after this identities quiz?
Re-derive any formula you missed from \(\sin(A+B)\). Then retry the quiz after a day and compare scores. Once you reach 9 of 10, attempt the verification proofs on the worksheet.
See every chapter on the Precalculus math page.
