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<h1 class="center">Proof of the Derivatives of<br>
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sin, cos and tan</h1>
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<p class="center">The three most useful derivatives in trigonometry are:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> sin(x) = cos(x)</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> cos(x) = −sin(x)</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> tan(x) = sec<sup>2</sup>(x)</p>
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Did they just drop out of the sky? Can we prove them somehow?
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<h2>Proving the Derivative of Sine</h2>
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<p>We need to go back, right back to first principles, the basic formula for derivatives:</p>
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<p class="center large"><span class="intbl"><em>dy</em><strong>dx</strong></span> = <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>f(x+Δx)−f(x)</em><strong>Δx</strong></span></p>
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<p>Pop in sin(x):</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>sin(x+Δx)−sin(x)</em><strong>Δx</strong></span></p>
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<p>We can then use this <a href="../algebra/trigonometric-identities.html">trigonometric identity</a>: sin(A+B) = sin(A)cos(B) + cos(A)sin(B) to get:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>sin(x)cos(Δx) + cos(x)sin(Δx) − sin(x)</em><strong>Δx</strong></span></p>
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<p>Regroup:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>sin(x)(cos(Δx)−1) + cos(x)sin(Δx)</em><strong>Δx</strong></span></p>
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<p>Split into two limits:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>sin(x)(cos(Δx)−1)</em><strong>Δx</strong></span> + <span class="lim"><em>lim</em><strong>Δx→0</strong></span><span class="intbl"><em>cos(x)sin(Δx)</em><strong>Δx</strong></span></p>
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<p>And we can bring sin(x) and cos(x) outside the limits because they are functions of x not Δx</p>
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<p class="center large">sin(x) <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(Δx)−1</em><strong>Δx</strong></span> + cos(x) <span class="lim"><em>lim</em><strong>Δx→0</strong></span><span class="intbl"><em> sin(Δx)</em><strong>Δx</strong></span></p>
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<p> </p>
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<p>Now all we have to do is evaluate those two little limits. Easy, right? Ha!</p>
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<h2>Limit of <span class="intbl"><em>sin(θ)</em><strong>θ</strong></span></h2>
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<p>Starting with</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>θ→0</strong></span> <span class="intbl"><em>sin(θ)</em><strong>θ</strong></span></p>
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<p>with the help of some geometry:</p>
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<p class="center"><img src="images/derivatives-trig1.svg" alt="circle with radius, angle and tangent"></p>
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<p>We can look at areas:</p>
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<p class="center larger">Area of triangle AOB <b><</b> Area of sector AOB <b><</b> Area of triangle AOC</p>
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<p class="center larger"><span class="intbl"><em>1</em><strong>2</strong></span>r<sup>2</sup> sin(θ) <b><</b> <span class="intbl"><em>1</em><strong>2</strong></span>r<sup>2</sup> θ <b><</b> <span class="intbl"><em>1</em><strong>2</strong></span>r<sup>2</sup> tan(θ)</p>
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<p>Divide all terms by <span class="intbl"><em>1</em><strong>2</strong></span>r<sup>2</sup> sin(θ)</p>
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<p class="center large">1 < <span class="intbl"><em>θ</em><strong>sin(θ)</strong></span> < <span class="intbl"><em>1</em><strong>cos(θ)</strong></span></p>
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<p>Take the reciprocals:</p>
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<p class="center large">1 > <span class="intbl"><em>sin(θ)</em><strong>θ </strong></span> > cos(θ)</p>
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<p>Now as θ→0 then cos(θ)→1</p>
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<p class="center larger">So <span class="intbl"><em>sin(θ)</em><strong>θ </strong></span> lies between 1 and something that is tending towards 1</p>
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<p>So as θ→0 then <span class="intbl"><em>sin(θ)</em><strong>θ </strong></span>→1 and so:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>θ→0</strong></span> <span class="intbl"><em>sin(θ)</em><strong>θ</strong></span> = 1</p>
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<p>(Note: we should also prove this is true from the negative side, how about you try with negative values of θ ?)</p>
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<h2>Limit of <span class="intbl"><em>cos(θ)−1</em><strong>θ</strong></span></h2>
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<p>So next we want to find out this one:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>θ→0</strong></span> <span class="intbl"><em>cos(θ)−1</em><strong>θ</strong></span></p>
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<p>When we multiply top and bottom by cos(θ)+1 we get:</p>
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<p class="center large"><span class="intbl"><em>(cos(θ)−1)(cos(θ)+1)</em><strong>θ(cos(θ)+1)</strong></span> = <span class="intbl"><em>cos<sup>2</sup>(θ)−1</em><strong>θ(cos(θ)+1)</strong></span></p>
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<p>Now we use this <a href="../algebra/trigonometric-identities.html">trigonometric identity</a> based on <a href="../pythagoras.html">Pythagoras' Theorem</a>:</p>
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<p class="center large">cos<sup>2</sup>(x) + sin<sup>2</sup>(x) = 1</p>
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<p>Rearranged to this form:</p>
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<p class="center large">cos<sup>2</sup>(x) − 1 = −sin<sup>2</sup>(x)</p>
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<p>And the limit we started with can become:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>θ→0</strong></span> <span class="intbl"><em>−sin<sup>2</sup>(θ)</em><strong>θ(cos(θ)+1)</strong></span></p>
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<p>That looks worse! But is really better because we can turn it into two limits multiplied together:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>θ→0</strong></span><span class="intbl"><em>sin(θ)</em><strong>θ</strong></span> × <span class="lim"><em>lim</em><strong>θ→0</strong></span><span class="intbl"><em>−sin(θ)</em><strong>cos(θ)+1</strong></span></p>
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<p>We know the first limit (we worked it out above), and the second limit doesn't need much work because<b> at θ=0</b> we know directly that <span class="intbl"><em>−sin(0)</em><strong>cos(0)+1</strong></span> = 0, so:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>θ→0</strong></span><span class="intbl"><em>sin(θ)</em><strong>θ</strong></span> × <span class="lim"><em>lim</em><strong>θ→0</strong></span><span class="intbl"><em>−sin(θ)</em><strong>cos(θ)+1</strong></span> = 1 × 0 = 0</p>
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<h2>Putting it Together</h2>
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<p>So what were we trying to do again? Oh that's right, we really wanted to work out this:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = sin(x) <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(Δx)−1</em><strong>Δx</strong></span> + cos(x) <span class="lim"><em>lim</em><strong>Δx→0</strong></span><span class="intbl"><em> sin(Δx)</em><strong>Δx</strong></span></p>
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<p>We can now put in the values we just worked out and get:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = sin(x) × 0 + cos(x) × 1</p>
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<p>And so (ta da!):</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = cos(x)</p>
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<h2>The Derivative of Cosine</h2>
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<p>Now on to cosine!</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>cos(x) = <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(x+Δx)−cos(x)</em><strong>Δx</strong></span></p>
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<p>This time we will use the <a href="../algebra/trigonometric-identities.html">angle formula</a> <span class="center"><b>cos(A+B) = cos(A)cos(B) − sin(A)sin(B)</b></span>:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(x)cos(Δx) − sin(x)sin(Δx) − cos(x)</em><strong>Δx</strong></span></p>
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<p>Rearrange to:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(x)(cos(Δx)−1) − sin(x)sin(Δx)</em><strong>Δx</strong></span></p>
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<p>Split into two limits:</p>
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<p class="center large"><span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(x)(cos(Δx)−1)</em><strong>Δx</strong></span> − <span class="lim"><em>lim</em><strong>Δx→0</strong></span><span class="intbl"><em>sin(x)sin(Δx)</em><strong>Δx</strong></span></p>
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<p>We can bring cos(x) and sin(x) outside the limits because they are functions of x not Δx</p>
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<p class="center large">cos(x) <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>cos(Δx)−1</em><strong>Δx</strong></span> − sin(x) <span class="lim"><em>lim</em><strong>Δx→0</strong></span><span class="intbl"><em> sin(Δx)</em><strong>Δx</strong></span></p>
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<p>And using our knowledge from above:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> cos(x) = cos(x) × 0 − sin(x) × 1</p>
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<p>And so:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> cos(x) = −sin(x)</p>
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<h2>The Derivative of Tangent</h2>
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<p>To find the derivative of tan(x) we can use this <a href="../algebra/trigonometric-identities.html">identity</a>:</p>
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<p class="center large">tan(x) = <span class="intbl"><em>sin(x)</em><strong>cos(x)</strong></span></p>
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<p>So we start with:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) = <span class="intbl"><em>d</em><strong>dx</strong></span>(<span class="intbl"><em>sin(x)</em><strong>cos(x)</strong></span>)</p>
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<div class="def">
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<p>Now we can use the <a href="derivatives-rules.html">quotient rule</a> of derivatives:</p>
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<p class="center larger">(<span class="intbl"><em>f</em><strong>g</strong></span>)’ = <span class="intbl"><em>gf’ − fg’</em><strong>g<sup>2</sup></strong></span></p>
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</div>
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<p>And we get:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) = <span class="intbl"><em>cos(x) × cos(x) − sin(x) × −sin(x)</em><strong>cos<sup>2</sup>(x)</strong></span></p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) = <span class="intbl"><em>cos<sup>2</sup>(x) + sin<sup>2</sup>(x)</em><strong>cos<sup>2</sup>(x)</strong></span></p>
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<p>Then use this identity:</p>
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<p class="center larger">cos<sup>2</sup>(x) + sin<sup>2</sup>(x) = 1</p>
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<p>To get</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) =<span class="intbl"><em>1</em><strong>cos<sup>2</sup>(x)</strong></span></p>
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<p>Done!</p>
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<p>But most people like to use the fact that cos = <span class="intbl"><em>1</em><strong>sec</strong></span> to get:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) = sec<sup>2</sup>(x)</p>
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<p>Note: we can also do this:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) = <span class="intbl"><em>cos<sup>2</sup>(x) + sin<sup>2</sup>(x)</em><strong>cos<sup>2</sup>(x)</strong></span></p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>tan(x) = 1 + <span class="intbl"><em> sin<sup>2</sup>(x)</em><strong>cos<sup>2</sup>(x)</strong></span> = 1 + tan<sup>2</sup>(x)</p>
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<p>(And, yes, 1 + tan<sup>2</sup>(x) = sec<sup>2</sup>(x) anyway, see <a href="../algebra/trig-magic-hexagon.html">Magic Hexagon</a> )</p>
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<p class="center"> </p>
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<h2>Taylor Series</h2>
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<p>Just on a fun side note, we can use the <a href="../algebra/taylor-series.html">Taylor Series</a> expansions and differentiate term by term.</p>
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<div class="example">
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<h3>Example: sin(x) and cos(x)</h3>
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<p>The Taylor Series expansion for sin(x) is</p>
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<p class="center large">sin(x) = x − <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>x<sup>5</sup></em><strong>5!</strong></span> − ...</p>
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<p>Differentiate term by term:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> sin(x) = 1 − <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> − ...</p>
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<p>Which perfectly matches the Taylor Series expansion for cos(x)</p>
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<p class="center large">cos(x) = 1 − <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> − ...</p>
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<p> </p>
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<p>Let's also differentiate <b>that</b> term by term:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span> cos(x) = 0 − x + <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span><strong> </strong>− ...</p>
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<p>Which is the <b>negative</b> of the Taylor Series expansion for sin(x) we started with!</p>
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</div>
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<p>But this is "circular reasoning" because the original expansion of the Taylor Series already use the rules "the derivative of sin(x) is cos(x)" and "the derivative of cos(x) is −sin(x)".</p>
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<p> </p>
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