M151Q Precalculus · Unit 4: Trigonometric functions · on Final Exam
TRIG 5.1, 5.3 Algebra with trig ratios and trigonometric identities
Study notes for Algebra with trig ratios and trigonometric identities in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Treat $\sin\theta$ and $\cos\theta$ like variables when simplifying: $\sin\theta\cos\theta + 2\sin\theta = \sin\theta(\cos\theta + 2)$. Careful: $\sin 2\theta \ne 2\sin\theta$ and $\sin^2\theta$ means $(\sin\theta)^2$.
- Fundamental identities: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$, $\sin^2\theta + \cos^2\theta = 1$ (and its forms $1 - \cos^2\theta = \sin^2\theta$), $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$.
- Prove an identity by transforming one side into the other (usually the messier side): convert everything to sine and cosine, combine fractions, factor, use the Pythagorean identity.
- Solve equations by isolating one trig function, using identities to reduce to a single function, then finding all angles in the interval.
Key formulas
Fundamental identities
$$\tan\theta = \frac{\sin\theta}{\cos\theta},\qquad \sin^2\theta + \cos^2\theta = 1,\qquad 1 + \tan^2\theta = \sec^2\theta$$
Negative angles
$$\sin(-\theta) = -\sin\theta,\quad \cos(-\theta) = \cos\theta,\quad \tan(-\theta) = -\tan\theta$$
Worked example
Simplify $\dfrac{1 - \cos^2\theta}{\sin\theta\cos\theta}$ and solve $2\sin^2\theta = 1$ for $0^\circ \le \theta < 360^\circ$.
Show the solution
$\dfrac{\sin^2\theta}{\sin\theta\cos\theta} = \dfrac{\sin\theta}{\cos\theta} = \tan\theta$. $\sin^2\theta = \tfrac12 \Rightarrow \sin\theta = \pm\tfrac{\sqrt2}{2} \Rightarrow \theta = 45^\circ, 135^\circ, 225^\circ, 315^\circ$.
Common mistakes
- Dividing an equation by $\sin\theta$ and losing the solutions where $\sin\theta = 0$.
- Writing $\sin\theta + \cos\theta = 1$ as an identity (only $\sin^2 + \cos^2 = 1$).
Exam tipWhen stuck on an identity, write everything in sines and cosines and combine into a single fraction. It almost always unlocks.
Keep going in Mathub
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