Establish and use the derivatives of sine, cosine and tangent functions, including the chain rule for trigonometric functions of a function of x

What this dot point is asking

SCSA Unit 3 requires fluency differentiating the three core trigonometric functions and their composites. These derivatives appear in both the calculator-free and calculator-assumed sections and combine constantly with the product, quotient and chain rules.

The standard derivatives

The derivatives below assume the angle is measured in radians. The simple form $\dfrac{d}{dx}\sin x=\cos x$ only holds because $\lim_{h\to0}\dfrac{\sin h}{h}=1$ when $h$ is in radians; in degrees an awkward constant factor appears.

Note the sign: differentiating cosine introduces a minus, while differentiating sine does not. The derivative of $\tan x$ follows from the quotient rule applied to $\dfrac{\sin x}{\cos x}$.

The chain rule with trigonometric functions

When the angle is a function of $x$, the chain rule multiplies by the derivative of that angle. For $y=\sin(kx)$ the angle $kx$ has derivative $k$, so $y'=k\cos(kx)$.

Combining with other rules

Trigonometric derivatives frequently sit inside a product or quotient. The strategy is to label each factor and differentiate it, applying the chain rule to any composite angle.

Why radians are required

All these derivative rules are only valid when $x$ is in radians. If a question is set in degrees you must convert, or differentiate $\sin\!\left(\dfrac{\pi x}{180}\right)$ with the chain rule, which produces the awkward factor $\dfrac{\pi}{180}$. SCSA exam questions involving calculus use radians.