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232 lines
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<title>Finding an Angle in a Right Angled Triangle</title>
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<h1 class="center">Finding an Angle in a Right Angled Triangle</h1>
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<h2>Angle from Any Two Sides</h2>
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<p>We can find an <b>unknown angle</b> in a <a href="../right_angle_triangle.html">right-angled triangle</a>, as long as we know the lengths of <b>two of its sides</b>.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-ladder.gif" alt="ladder against wall" height="179" width="224" ></p>
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<h3>Example</h3>
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<p>The ladder leans against a wall as shown.</p>
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<p>What is the <b>angle</b> between the ladder and the wall?</p><p> </p>
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</div>
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<div style="clear:both"></div>
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<p class="center large">The answer is to use <a href="../sine-cosine-tangent.html">Sine, Cosine or Tangent</a>!</p>
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<p>But which one to use? We have a special phrase "<a href="sohcahtoa.html">SOHCAHTOA</a>" to help us, and we use it like this:</p>
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<div class="dotpoint">
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<p><b>Step 1</b>: find the <b>names</b> of the two sides we know</p>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/adjacent-opposite-hypotenuse.svg" alt="triangle showing Opposite, Adjacent and Hypotenuse" height="190" width="326" ></p>
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<ul>
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<li><b>Adjacent</b> is adjacent to the angle,</li>
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<li><b>Opposite</b> is opposite the angle,</li>
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<li>and the longest side is the <b>Hypotenuse</b>.</li>
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</ul>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: in our ladder example we know the length of:</h3>
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<ul>
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<li>the side <b>Opposite</b> the angle "x", which is <b>2.5</b></li>
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<li>the longest side, called the <b>Hypotenuse</b>, which is <b>5</b></li>
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</ul>
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</div>
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<div class="dotpoint">
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<p><b>Step 2</b>: now use the first letters of those two sides (<b>O</b>pposite and <b>H</b>ypotenuse) and the phrase "<a href="sohcahtoa.html">SOHCAHTOA</a>" to find which one of Sine, Cosine <b>or</b> Tangent to use:</p>
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</div>
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<table style="border: 0; margin:auto;">
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<tr>
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<td style="width:80px;"><div align="left"><b><i>SOH...</i></b></div></td>
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<td style="width:350px;"><div class="center"><b>S</b>ine: sin(θ) = <b>O</b>pposite / <b>H</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td style="width:80px;"><div class="center"><b><i>...CAH...</i></b></div></td>
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<td style="width:350px;"><div class="center"><b>C</b>osine: cos(θ) = <b>A</b>djacent / <b>H</b>ypotenuse</div></td>
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</tr>
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<tr>
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<td style="width:80px;"><div align="right"><b><i>...TOA</i></b></div></td>
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<td style="width:350px;"><div class="center"><b>T</b>angent: tan(θ) = <b>O</b>pposite / <b>A</b>djacent</div></td>
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</tr>
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</table>
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<div class="example">
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<p>In our example that is<b> O</b>pposite and<b> H</b>ypotenuse, and that gives us “<b>SOH</b>cahtoa”, which tells us we need to use <b>Sine</b>.</p>
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</div>
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<div class="dotpoint">
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<p><b>Step 3</b>: Put our values into the Sine equation:</p>
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<p class="center larger"><b>S</b>in (x) = <b>O</b>pposite / <b>H</b>ypotenuse = 2.5 / 5 = <b>0.5</b></p>
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</div>
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<div class="dotpoint">
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<p><b>Step 4</b>: Now solve that equation!</p>
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<p class="center larger">sin(x) = 0.5</p>
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<p>Next (trust me for the moment) we can re-arrange that into this:</p>
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<p class="center larger">x = sin<sup>-1</sup>(0.5)</p>
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<p>And then get our calculator, key in 0.5 and use the sin<sup>-1</sup> button to get the answer:</p>
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<p class="center larger">x = <b>30°</b></p>
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</div>
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And we have our answer!
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<div class="def">
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<h3>But what is the meaning of <b>sin<sup>-1</sup></b> … ?</h3>
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<p class="center">Well, the Sine function <i><b>"sin"</b></i> takes an angle and gives us the <b>ratio</b> "opposite/hypotenuse",</p>
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<div class="center">
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<p><img src="images/sin-sin-1.svg" alt="sin vs sin-1" height="131" width="338" ></p>
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<p>But <i><b>sin<sup>-1</sup></b></i> (called "inverse sine") goes the other way ...<br>
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... it
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takes the <b>ratio</b> "opposite/hypotenuse" and gives us an angle.</p>
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</div>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<ul>
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<li>Sine Function: sin(<b>30°</b>) = <b>0.5</b></li>
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<li>Inverse Sine Function: sin<sup>-1</sup>(<b>0.5</b>) = <b>30°</b></li>
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</ul>
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</div>
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<p> </p>
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<table style="border: 0; margin:auto;">
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<tr>
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<td><img src="images/calculator-sin-cos-tan.jpg" alt="calculator-sin-cos-tan" height="75" width="118" ></td>
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<td>On the calculator press one of the following (depending<br>
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on your brand of calculator):
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either '2ndF sin' or 'shift sin'.</td>
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</tr>
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</table>
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<p class="center">On your calculator, try using <b>sin</b> and <b>sin<sup>-1</sup></b> to see what results you get!</p>
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<p class="center">Also try <b>cos</b> and <b>cos<sup>-1</sup></b>. And <b>tan</b> and <b>tan<sup>-1</sup></b>.<br>
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Go on, have a try now.</p>
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<h2>Step By Step</h2>
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<p>These are the four steps we need to follow:</p><ul>
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<li><b>Step 1</b> Find which two sides we know – out of Opposite, Adjacent and Hypotenuse.</li><li><b>Step 2</b> Use SOHCAHTOA to decide which one of Sine, Cosine <b>or</b> Tangent to use in this question.</li>
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<li><b>Step 3</b> For Sine calculate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse <b>or</b> for Tangent calculate Opposite/Adjacent.</li><li><b>Step 4</b> Find the angle from your calculator, using one of sin<sup>-1</sup>, cos<sup>-1</sup> <b>or</b> tan<sup>-1</sup></li></ul>
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<h2>Examples</h2>
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<p>Let’s look at a couple more examples:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-example1.gif" alt="trig example airplane 400, 300" height="160" width="220" ></p>
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<h3>Example</h3>
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<p>Find the angle of elevation
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of the plane from point A on the ground.</p>
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<div style="clear:both"></div><br>
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<ul>
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<li><b>Step 1</b> The two sides we know are <b>O</b>pposite (300) and <b>A</b>djacent (400).</li><li><b>Step 2</b> SOHCAH<b>TOA</b> tells us we must use <b>T</b>angent.</li>
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<li><b>Step 3</b> Calculate <b>Opposite/Adjacent</b> = 300/400 = <b>0.75</b></li>
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<li><b>Step 4</b> Find the angle from your calculator using <b>tan<sup>-1</sup></b></li>
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</ul>
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<p class="indent50px">Tan x° = opposite/adjacent = 300/400 = 0.75</p>
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<p class="indent50px"><b>tan<sup>-1</sup></b> of 0.75 = <b>36.9°</b> (correct to 1 decimal place)</p>
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</div>
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<p>Unless you’re told otherwise, angles are usually rounded to one place of decimals.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/trig-example2.gif" alt="trig example" height="124" width="194" ></p>
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<h3>Example</h3>
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<p>Find the size of angle a°</p>
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<div style="clear:both"></div><br>
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<ul>
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<li><b>Step 1</b> The two sides we know are <b>A</b>djacent (6,750) and <b>H</b>ypotenuse (8,100).</li><li><b>Step 2</b> SOH<b>CAH</b>TOA tells us we must use <b>C</b>osine.</li>
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<li><b>Step 3</b> Calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333</li>
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<li><b>Step 4</b> Find the angle from your calculator using <b>cos<sup>-1</sup></b> of 0.8333:</li>
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</ul><div class="indent50px">
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<p>cos a° = 6,750/8,100 = 0.8333</p>
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</div>
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<div class="indent50px"> <b>cos<sup>-1</sup></b> of 0.8333 = <b>33.6°</b> (to 1 decimal place)</div>
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</div>
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<p> </p>
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<div class="questions">250, 1500, 1501, 1502, 251, 1503, 2349, 2350, 2351, 3934</div>
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<div class="related">
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<a href="../games/random-trigonometry.html">Random Trigonometry</a>
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<a href="trig-sine-law.html">The Law of Sines</a>
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<a href="trig-cosine-law.html">The Law of Cosines</a>
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<a href="trig-solving-triangles.html">Solving Triangles</a>
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<a href="trigonometry-index.html">Trigonometry Index</a>
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<a href="index.html">Algebra Index</a>
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