M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§2.2 Sine and cosine
Study notes for Sine and cosine in M171 Calculus I 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 StudyCat.
Big ideas
- $\frac{d}{dx}[\sin x] = \cos x$ and $\frac{d}{dx}[\cos x] = -\sin x$. These are true when $x$ is in radians.
- Remember the cycle: $\sin \to \cos \to -\sin \to -\cos \to \sin$.
- Know the unit-circle values at $0,\ \frac\pi6,\ \frac\pi4,\ \frac\pi3,\ \frac\pi2,\ \pi$ so you can evaluate $f'$ at specific points without a calculator.
Key formulas
Sine and cosine
$$\frac{d}{dx}[\sin x] = \cos x,\qquad \frac{d}{dx}[\cos x] = -\sin x$$
Worked example
Find the tangent line to $f(x) = 3\sin x - 2\cos x$ at $x = 0$.
Show the solution
$f(0) = -2$, $f'(x) = 3\cos x + 2\sin x$, $f'(0) = 3$. Tangent line: $y = -2 + 3x$.
Common mistakes
- Sign error on $\frac{d}{dx}[\cos x]$.
- Using degrees. All calculus with trig functions is in radians.
Exam tipSketch the unit circle in the margin on exam day: it is faster than trying to recall $\cos\frac{2\pi}{3}$ from memory.
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