Pythagorean Trig Equations | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Divide the familiar identity by the required square

An identity is true for every input where both sides are defined. The sign $\equiv$ emphasises this. Starting with $\sin^2x+\cos^2x=1$ , divide every term by $\cos^2x$ , provided cosine is non-zero.

\begin{gathered}\frac{\sin^2x}{\cos^2x}+1=\frac1{\cos^2x}\\ \tan^2x+1\equiv\sec^2x\end{gathered}

Dividing by $\sin^2x$ instead gives $1+\cot^2x\equiv\cosec^2x$ , where sine is non-zero. Here $\sec^2x$ means $(\sec x)^2$ , not $\sec(2x)$ . These identities are supplied in the MF19 formula booklet.

Choose the identity that leaves only one ratio. For example, with an unsquared secant term, replace $\tan^2x$ by $\sec^2x-1$ ; with an unsquared tangent term, replace secant squared instead.

Your turnPractice

Rewrite $3\sec^2x-2\tan x=4$ using tangent only. Which angles must still be excluded?

Show worked answer

Answer

3\tan^2x-2\tan x-1=0

Exclude every angle with $\cos x=0$ ; neither tangent nor secant is defined there.

Solve the algebra before finding the angles

Worked example

Solve 2 sec²θ − 5 sec θ + 2 = 0 for −π ≤ θ ≤ π.

Let $u=\sec\theta$ . A polynomial calculator quickly finds the two algebraic roots: choose a degree-2 polynomial and enter coefficients 2, −5, 2. A root $r$ gives a factor $u-r$ ; keep the original leading coefficient too. The roots are convenient here, so write and expand-check the matching factors.

\begin{gathered}2u^2-5u+2=0\\(2u-1)(u-2)=0\\ u=\frac12\quad\text{or}\quad u=2\end{gathered}

Reject $u=1/2$ because secant has magnitude at least 1. The other root gives $\cos\theta=1/2$ .

Answer

\theta=-\frac\pi3,\ \frac\pi3

Both original secant values are 2. If calculator roots are awkward and exact working is needed, use the quadratic formula before converting to angles.

Your turnPractice

Solve $\sec^2x-\tan x-3=0$ for $0^\circ\le x\le360^\circ$ .

Show worked answer

\begin{gathered}u=\tan x\\u^2-u-2=0\\(u-2)(u+1)=0\end{gathered}

Tangent 2 gives $x\approx63.4^\circ,243.4^\circ$ . Tangent −1 gives $135^\circ,315^\circ$ . All four have non-zero cosine. Round non-exact degree answers to one decimal place unless instructed otherwise.

Your turnPractice

Solve $2\cot^2x-3\cosec x=0$ for $0<x<2\pi$ .

Show worked answer

\begin{gathered}u=\cosec x\\2u^2-3u-2=0\\(2u+1)(u-2)=0\end{gathered}

Reject cosecant $-1/2$ . Cosecant 2 gives sine $1/2$ .

Answer

x=\frac\pi6,\ \frac{5\pi}6

Divide the familiar identity by the required square

\begin{gathered}\frac{\sin^2x}{\cos^2x}+1=\frac1{\cos^2x}\\ \tan^2x+1\equiv\sec^2x\end{gathered}

Choose the identity that leaves only one ratio. For example, with an unsquared secant term, replace $\tan^2x$ by $\sec^2x-1$ ; with an unsquared tangent term, replace secant squared instead.

Your turnPractice

Rewrite $3\sec^2x-2\tan x=4$ using tangent only. Which angles must still be excluded?

Show worked answer

Answer

3\tan^2x-2\tan x-1=0

Exclude every angle with $\cos x=0$ ; neither tangent nor secant is defined there.

Solve the algebra before finding the angles

Worked example

Solve 2 sec²θ − 5 sec θ + 2 = 0 for −π ≤ θ ≤ π.

\begin{gathered}2u^2-5u+2=0\\(2u-1)(u-2)=0\\ u=\frac12\quad\text{or}\quad u=2\end{gathered}

Reject $u=1/2$ because secant has magnitude at least 1. The other root gives $\cos\theta=1/2$ .

Answer

\theta=-\frac\pi3,\ \frac\pi3

Both original secant values are 2. If calculator roots are awkward and exact working is needed, use the quadratic formula before converting to angles.

Your turnPractice

Solve $\sec^2x-\tan x-3=0$ for $0^\circ\le x\le360^\circ$ .

Show worked answer

\begin{gathered}u=\tan x\\u^2-u-2=0\\(u-2)(u+1)=0\end{gathered}

Your turnPractice

Solve $2\cot^2x-3\cosec x=0$ for $0<x<2\pi$ .

Show worked answer

\begin{gathered}u=\cosec x\\2u^2-3u-2=0\\(2u+1)(u-2)=0\end{gathered}

Reject cosecant $-1/2$ . Cosecant 2 gives sine $1/2$ .

Answer

x=\frac\pi6,\ \frac{5\pi}6