 New Zealand
Level 8 - NCEA Level 3

Tangent Ratios (proofs and equivalences)

Lesson

Imagine a right-angled triangle with one of the acute angles labelled $\phi$ϕ. If the side opposite the angle $\phi$ϕ has length $a$a and the side adjacent to the angle $\phi$ϕ has length $b$b, then we write $\tan\phi=\frac{a}{b}$tanϕ=ab. But this is the same as the equivalent fraction  $\frac{\frac{a}{c}}{\frac{b}{c}}$acbc obtained by dividing the numerator and the denominator by the number $c$c.

If we now let $c$c be the hypotenuse of the triangle, we see that the complicated looking fraction is just $\frac{\sin\phi}{\cos\phi}$sinϕcosϕ. Hence, we can write the identity

$\tan\phi\equiv\frac{\sin\phi}{\cos\phi}$tanϕsinϕcosϕ

(This is an identity in the sense that it is true for every value of $\phi$ϕ.)

Another very useful identity is obtained in a similar way. In the same triangle, we have the statement of Pythagoras's theorem $a^2+b^2\equiv c^2$a2+b2c2, which remains true for a given $c$c whatever the values of $a$a and $b$b, provided the triangle continues to be right-angled.

On dividing both sides of this equivalence by $c^2$c2, we obtain $\frac{a^2}{c^2}+\frac{b^2}{c^2}\equiv1$a2c2+b2c21. We recognise this as just

$\sin^2\phi+\cos^2\phi\equiv1$sin2ϕ+cos2ϕ1.

This can now be divided through by $\cos^2\phi$cos2ϕ to obtain a further identity involving the tangent function:

$\tan^2\phi+1\equiv\frac{1}{\cos^2\phi}=\sec^2\phi$tan2ϕ+11cos2ϕ=sec2ϕ

Examples

Example 1

Find a simpler way of writing $\frac{\sin^2A+\sin A\cos A}{\cos^2A+\sin A\cos A}$sin2A+sinAcosAcos2A+sinAcosA.

Taking a common factor out of the numerator and out of the denominator, we can write the expression as

$\frac{\sin A\left(\sin A+\cos A\right)}{\cos A\left(\cos A+\sin A\right)}$sinA(sinA+cosA)cosA(cosA+sinA).

On cancelling the common term from the numerator and the denominator, we have $\frac{\sin A}{\cos A}=\tan A$sinAcosA=tanA.

Often, students are asked to verify an identity. This is usually done by rearranging or simplifying one side of the identity until it looks identical to the other side. The working above essentially proves the identity

$\frac{\sin^2A+\sin A\cos A}{\cos^2A+\sin A\cos A}\equiv\tan A$sin2A+sinAcosAcos2A+sinAcosAtanA.

Example 2

If $5\sin x=13\cos x$5sinx=13cosx and $x$x is between $0$0 and $2\pi$2π, what are the possible values of $x$x?

From $5\sin x=13\cos x$5sinx=13cosx we obtain $\frac{\sin x}{\cos x}=\frac{13}{5}=2.6$sinxcosx=135=2.6

That is, $\tan x=2.6$tanx=2.6. Therefore, in the first quadrant, $x=\arctan2.6\approx1.2036$x=arctan2.61.2036 or $\frac{\pi}{2.6101}$π2.6101.  There should also be a third-quadrant solution since $\tan x=\tan\left(\pi+x\right)$tanx=tan(π+x). That is, the second solution is$\pi+\frac{\pi}{2.6101}=\frac{3.6101\times\pi}{2.6101}$π+π2.6101=3.6101×π2.6101.

Worked Examples

Question 1

By finding the ratio represented by $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ in the given figure, we want to prove that $\frac{\sin\theta}{\cos\theta}=\tan\theta$sinθcosθ=tanθ. 1. Write down the expression for $\sin\theta$sinθ.

2. Write down the expression for $\cos\theta$cosθ.

3. Hence, form an expression for $\frac{\sin\theta}{\cos\theta}$sinθcosθ.

4. Write down the expression for $\tan\theta$tanθ.

5. Does $\frac{\sin\theta}{\cos\theta}=\tan\theta$sinθcosθ=tanθ?

Yes

A

No

B

Yes

A

No

B

Question 2

.

Given that $\sin x=\frac{4}{5}$sinx=45 and $\cos x=\frac{3}{5}$cosx=35, find $\tan x$tanx.

Question 3

Find the value of $x$x when

$8\sin x=15\cos x$8sinx=15cosx.

Outcomes

M8-6

Manipulate trigonometric expressions

91575

Apply trigonometric methods in solving problems