Trigonometry

Explanation

• For a unit-circle angle $\theta$, $\cos\theta$ and $\sin\theta$ are the point's horizontal and vertical coordinates, while $\tan\theta=\sin\theta/\cos\theta$ where $\cos\theta\neq0$; this extends the ratios beyond acute angles.
• For a triangle, use $a/\sin A=b/\sin B=c/\sin C$ when an opposite side-angle pair is known, $a^2=b^2+c^2-2bc\cos A$ for three sides or an included angle, and $\tfrac12ab\sin C$ for area.
• In the ambiguous sine-rule case, test the supplementary angle because $\sin A=\sin(\pi-A)$, then reject any triangle inconsistent with the given sides and angles.
• Radian measure makes arc and sector formulae direct: an angle $\theta$ radians in a circle of radius $r$ gives arc length $s=r\theta$ and sector area $A=\tfrac12r^2\theta$.
• Keep the calculator in the required angle mode and label sides opposite their matching angles; using degrees in $s=r\theta$ or pairing the wrong side and angle is a common error.

Explanation

• For $|\theta|$ close to zero and measured in radians, $\sin\theta\approx\theta$, $\tan\theta\approx\theta$ and $\cos\theta\approx1-\theta^2/2$. Replace each trigonometric function by its stated approximation, simplify algebraically, and retain only a solution whose magnitude is small enough for the approximation to be credible.
• For example, $1-\cos\theta\approx\theta^2/2$, so $(1-\cos\theta)/\theta^2\approx1/2$ for a small non-zero $\theta$.
• The approximations are radian results, not degree results; another common error is to accept a large root created by the approximate polynomial.
• The approximation sign matters: these are local approximations near zero, not identities.
• Quote the approximation used, substitute the small radian angle, and reject any root whose magnitude is inconsistent with the small-angle assumption.

Explanation

• $\sin x$ and $\cos x$ have range $[-1,1]$ and period $2\pi$, while $\tan x$ has range $\mathbb{R}$, period $\pi$ and vertical asymptotes at $x=\pi/2+k\pi$.
• Use $\sin(-x)=-\sin x$, $\cos(-x)=\cos x$ and $\tan(-x)=-\tan x$, then reduce an angle by a whole period before using its reference angle and quadrant.
• Know exact sine and cosine values at $0$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$ and $\pi$, and exact tangent values where defined.
• Transformations such as $y=\cos(x+\pi/6)$ shift the graph, while $y=\tan2x$ halves its period.
• In $a\cos(bx)+c$, the amplitude is $|a|$, period $2\pi/|b|$, and $c$ moves the midline.

Explanation

• The reciprocal functions are $\sec x=1/\cos x$, $\cosec x=1/\sin x$ and $\cot x=\cos x/\sin x$; their graphs inherit zeros of the denominator as vertical asymptotes. Secant and cosecant have range $(-\infty,-1]\cup[1,\infty)$ and period $2\pi$, while cotangent has range $\mathbb{R}$ and period $\pi$.
• The inverse graphs are restrictions reflected in $y=x$.
• Their principal ranges are $-\pi/2\leq\arcsin x\leq\pi/2$, $0\leq\arccos x\leq\pi$ and $-\pi/2<\arctan x<\pi/2$; $\arcsin x$ and $\arccos x$ require $-1\leq x\leq1$.
• Angles may be in degrees or radians.
• The notation $\sin^{-1}x$ means $\arcsin x$, not the reciprocal $\cosec x$.

Explanation

• The quotient identity is $\tan\theta=\sin\theta/\cos\theta$, and the three Pythagorean identities are $\sin^2\theta+\cos^2\theta=1$, $\sec^2\theta=1+\tan^2\theta$ and $\cosec^2\theta=1+\cot^2\theta$.
• Choose an identity containing the known and required functions, rearrange it, and use the stated quadrant to select the correct sign after taking a square root.
• If $\tan\theta=-7/24$ in quadrant II, a reference triangle has side magnitudes $7$, $24$ and $25$, so $\sin\theta=7/25$ and $\cos\theta=-24/25$.
• From $\sin^2\theta+\cos^2\theta=1$ one obtains two possible signs; ignoring the quadrant or silently choosing the positive root can change every reciprocal ratio that follows.

Explanation

• The compound-angle formulae are $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$, $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$ and $\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}$; note the sign reversal in cosine.
• Putting $A=B$ gives $\sin2\theta=2\sin\theta\cos\theta$, $\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$ and $\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$, and rearranging the cosine forms gives $\cos^2\theta=\frac12(1+\cos2\theta)$ for integration.
• To write $a\cos\theta+b\sin\theta$ as $R\cos(\theta\pm\alpha)$, expand the target, match coefficients to get $R\cos\alpha=a$ and $R\sin\alpha=b$, so $R=\sqrt{a^2+b^2}$ and $\tan\alpha=\dfrac ba$ with $\alpha$ in the stated range and in the stated units.
• The $R$ form then gives the maximum $R$, the minimum $-R$, the values of $\theta$ where they occur, and solves $a\cos\theta+b\sin\theta=c$ in one step.

Explanation

• Rearrange to a single ratio, find the principal value from the calculator, then use the CAST diagram or the graph to generate every solution in the interval: for $\cos$ the second solution is $360^\circ-\theta$ or $2\pi-\theta$, for $\sin$ it is $180^\circ-\theta$, for $\tan$ add $180^\circ$.
• A quadratic in one ratio is factorised or solved by the formula, and any root outside $[-1,1]$ for $\sin$ or $\cos$ is rejected before solving each branch; a Pythagorean identity converts a mixed equation into a quadratic in one ratio first.
• For a multiple angle, transform the interval, so $0<x<90^\circ$ becomes $0<2x<180^\circ$, find every value of $2x$ in that range, and divide at the end.
• Never divide by $\sin\theta$ or $\cos\theta$, which loses solutions; factorise instead.
• Check the mode, degrees or radians, against the interval, and check endpoints against strict or non-strict inequalities.

Explanation

• A trigonometric identity is true for every value in its domain, so a proof transforms one side into the other using exact algebra and established identities rather than testing selected angles.
• Usually begin with the more complicated side, replace secant, cosecant, cotangent or tangent by sine and cosine when helpful, and factor or take a common denominator before cancelling.
• Multiplying numerator and denominator by a conjugate can expose $1-\sin^2x=\cos^2x$ or $1-\cos^2x=\sin^2x$ and complete the proof cleanly.
• Never cancel terms across addition, and record domain restrictions: algebra such as division by $\sin x$ is valid only where that denominator is non-zero.

Explanation

• Resolve a vector of magnitude $V$ at angle $\theta$ to the positive horizontal into components $V\cos\theta$ and $V\sin\theta$, changing signs to match its actual direction.
• For resultant or equilibrium problems, form separate equations in two perpendicular directions; a zero resultant requires both component sums to equal zero.
• Trigonometric models can describe wave motion, a point on a vertical circular wheel or changing hours of sunlight.
• Interpret every solution using the stated time interval, units and physical constraints.
• A calculator angle without a quadrant check can point in the opposite direction; draw and label a diagram, then state bearings or directions in the form the context requests.