Trigonometry | Free ACT Math Course | The School of Mathematics

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<h1>ACT Math: Trigonometry</h1>

Trigonometry on the ACT goes a step beyond just SOH-CAH-TOA, roughly half of ACT trig questions draw on the Law of Sines and Cosines, radians and the unit circle, trig identities, and the graphs of sine and cosine. This free, complete lesson covers SOH-CAH-TOA and the Pythagorean identity, special angle values, radians and the unit circle, the Law of Sines and Law of Cosines, and graphs of sine and cosine (amplitude and period). Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

</p>

The ACT never requires inverse trig calculations to find an actual angle measure in degrees, questions stop at setting up or simplifying a ratio like "cos x = 4/5." Also, remember the ACT provides no formula sheet, so these formulas need to be memorized, and if using a calculator for a trig calculation, double-check it's set to degree mode unless a question specifically uses radians.

</div>

<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-trigonometry-quiz-1">Practice Trigonometry Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#soh-cah-toa">SOH-CAH-TOA and the Pythagorean Identity</a></li>

<li><a href="#special-angles">Special Angle Values</a></li>

<li><a href="#radians">Radians and the Unit Circle</a></li>

<li><a href="#law-sines-cosines">Law of Sines and Law of Cosines</a></li>

<li><a href="#graphs">Graphs of Sine and Cosine</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="soh-cah-toa">1. SOH-CAH-TOA and the Pythagorean Identity</h2>

<p class="step-math" style="text-align:center; font-size:1.05rem;">sin θ = <span class="frac"><span class="num">opposite</span><span class="den">hypotenuse</span></span>   cos θ = <span class="frac"><span class="num">adjacent</span><span class="den">hypotenuse</span></span>   tan θ = <span class="frac"><span class="num">opposite</span><span class="den">adjacent</span></span></p>

<p>For any angle θ, the <strong>Pythagorean identity</strong> holds:</p>

<p><strong>Worked Example:</strong> In a right triangle, angle θ has opposite side 6, adjacent side 8, hypotenuse 10. Find sin θ, cos θ, and tan θ.</p>

</div>

<p class="prompt">1. In a right triangle, angle θ has an opposite side of 9 and a hypotenuse of 15. Find sin θ.</p>

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<p><span class="label">Explanation:</span> sin θ = <span class="frac"><span class="num">9</span><span class="den">15</span></span> = <span class="frac"><span class="num">3</span><span class="den">5</span></span>.</p>

</div>

<p class="prompt">2. In a right triangle, angle θ has an adjacent side of 12 and a hypotenuse of 13. Find cos θ.</p>

</div>

<p><span class="label">Explanation:</span> cos θ = <span class="frac"><span class="num">12</span><span class="den">13</span></span>.</p>

</div>

<p class="prompt">3. In a right triangle, angle θ has an opposite side of 7 and an adjacent side of 24. Find tan θ.</p>

</div>

<p><span class="label">Explanation:</span> tan θ = <span class="frac"><span class="num">7</span><span class="den">24</span></span>.</p>

</div>

<p class="prompt">4. If sin θ = <span class="frac"><span class="num">3</span><span class="den">5</span></span>, and θ is acute, find cos θ using the Pythagorean identity.</p>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">9</span><span class="den">25</span></span> + cos² θ = 1, so cos² θ = <span class="frac"><span class="num">16</span><span class="den">25</span></span>, giving cos θ = <span class="frac"><span class="num">4</span><span class="den">5</span></span>.</p>

</div>

<p class="prompt">5. If cos θ = <span class="frac"><span class="num">5</span><span class="den">13</span></span>, and θ is acute, find sin θ using the Pythagorean identity.</p>

</div>

<p><span class="label">Explanation:</span> sin² θ + <span class="frac"><span class="num">25</span><span class="den">169</span></span> = 1, so sin² θ = <span class="frac"><span class="num">144</span><span class="den">169</span></span>, giving sin θ = <span class="frac"><span class="num">12</span><span class="den">13</span></span>.</p>

</div>

<h2 id="special-angles">2. Special Angle Values</h2>

<p>The sine, cosine, and tangent of 30°, 45°, and 60° come directly from the special right triangles, and are worth memorizing outright.</p>

</table>

<p><strong>Worked Example:</strong> Find sin(30°), cos(30°), and tan(30°) using the 30-60-90 triangle (side ratio 1 : √3 : 2).</p>

<p>For the 30° angle: opposite = 1, adjacent = √3, hypotenuse = 2.</p>

</div>

</div>

<p><span class="label">Explanation:</span> A 45-45-90 triangle has legs 1 and hypotenuse √2: sin(45°) = <span class="frac"><span class="num">1</span><span class="den">√2</span></span> = <span class="frac"><span class="num">√2</span><span class="den">2</span></span>.</p>

</div>

</div>

<p><span class="label">Explanation:</span> For the 60° angle in a 30-60-90 triangle, the adjacent side is 1 and the hypotenuse is 2: cos(60°) = <span class="frac"><span class="num">1</span><span class="den">2</span></span>.</p>

</div>

</div>

<p><span class="label">Explanation:</span> Since the two legs of a 45-45-90 triangle are equal, tan(45°) = <span class="frac"><span class="num">opposite</span><span class="den">adjacent</span></span> = 1.</p>

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<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-4',false)">D) Undefined</button>

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<p><span class="label">Explanation:</span> sin(90°) = 1, the maximum possible value of sine.</p>

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<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pb-5',false)">D) Undefined</button>

</div>

<p><span class="label">Explanation:</span> cos(0°) = 1, the maximum possible value of cosine.</p>

</div>

<h2 id="radians">3. Radians and the Unit Circle</h2>

<p>A full circle is 360°, which equals 2π radians. The <strong>unit circle</strong>, a circle of radius 1 centered at the origin, defines sine and cosine for any angle as the y- and x-coordinates of the point where that angle meets the circle.</p>

<p class="step-math" style="text-align:center;">radians = degrees × <span class="frac"><span class="num">π</span><span class="den">180</span></span>       degrees = radians × <span class="frac"><span class="num">180</span><span class="den">π</span></span></p>

<p><strong>Worked Example:</strong> Convert 120° to radians.</p>

</div>

<p class="prompt">1. Convert 270° to radians.</p>

</div>

<p><span class="label">Explanation:</span> 270 × <span class="frac"><span class="num">π</span><span class="den">180</span></span> = <span class="frac"><span class="num">3π</span><span class="den">2</span></span>.</p>

</div>

<p class="prompt">2. Convert <span class="frac"><span class="num">5π</span><span class="den">6</span></span> radians to degrees.</p>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">5π</span><span class="den">6</span></span> × <span class="frac"><span class="num">180</span><span class="den">π</span></span> = 150°.</p>

</div>

<p class="prompt">3. On the unit circle, what is the value of cos(0)?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-3',false)">D) Undefined</button>

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<p><span class="label">Explanation:</span> An angle of 0 corresponds to the point (1, 0) on the unit circle, so cos(0) = 1 (the x-coordinate).</p>

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<p class="prompt">4. On the unit circle, what is the value of sin(<span class="frac"><span class="num">π</span><span class="den">2</span></span>)?</p>

</div>

<p><span class="label">Explanation:</span> An angle of <span class="frac"><span class="num">π</span><span class="den">2</span></span> (90°) corresponds to the point (0, 1) on the unit circle, so sin(<span class="frac"><span class="num">π</span><span class="den">2</span></span>) = 1 (the y-coordinate).</p>

</div>

<p class="prompt">5. In which quadrant is an angle of <span class="frac"><span class="num">5π</span><span class="den">4</span></span> radians located?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pc-5',true)">A) Quadrant III</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">B) Quadrant I</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">C) Quadrant II</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-5',false)">D) Quadrant IV</button>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">5π</span><span class="den">4</span></span> radians equals 225°, which falls between 180° and 270°, placing it in Quadrant III.</p>

</div>

<h2 id="law-sines-cosines">4. Law of Sines and Law of Cosines</h2>

<p>For any triangle (not just right triangles) with sides a, b, c opposite angles A, B, C:</p>

<p class="step-math" style="text-align:center;">Law of Sines: <span class="frac"><span class="num">a</span><span class="den">sin A</span></span> = <span class="frac"><span class="num">b</span><span class="den">sin B</span></span> = <span class="frac"><span class="num">c</span><span class="den">sin C</span></span></p>

<p class="step-math" style="text-align:center;">Law of Cosines: a² = b² + c² − 2bc·cos A</p>

<p class="prompt">1. Which of the following correctly represents the Law of Sines for triangle ABC, with sides a, b, c opposite angles A, B, C respectively?</p>

</div>

<p><span class="label">Explanation:</span> The Law of Sines pairs each side with the sine of its own opposite angle: <span class="frac"><span class="num">a</span><span class="den">sin A</span></span> = <span class="frac"><span class="num">b</span><span class="den">sin B</span></span> = <span class="frac"><span class="num">c</span><span class="den">sin C</span></span>.</p>

</div>

<p class="prompt">2. Which of the following correctly represents the Law of Cosines for finding side a, opposite angle A?</p>

</div>

<p><span class="label">Explanation:</span> The Law of Cosines is a² = b² + c² − 2bc·cos A, note the minus sign before the last term.</p>

</div>

<p class="prompt">3. In a triangle, angle A = 30°, side a = 5, and angle B = 90°. Using the Law of Sines, what is the length of side b?</p>

</div>

<p><span class="label">Explanation:</span> <span class="frac"><span class="num">5</span><span class="den">sin 30°</span></span> = <span class="frac"><span class="num">b</span><span class="den">sin 90°</span></span>. Since sin 30° = 0.5 and sin 90° = 1: <span class="frac"><span class="num">5</span><span class="den">0.5</span></span> = 10 = <span class="frac"><span class="num">b</span><span class="den">1</span></span>, so b = 10.</p>

</div>

<p class="prompt">4. In triangle ABC, sides b = 7, c = 9, and angle A = 60°. Using the Law of Cosines, what is a² (before taking the square root)?</p>

</div>

<p><span class="label">Explanation:</span> a² = 7² + 9² − 2(7)(9)cos(60°) = 49 + 81 − 126(0.5) = 130 − 63 = 67.</p>

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<h2 id="graphs">5. Graphs of Sine and Cosine</h2>

<p>The graphs of y = sin(x) and y = cos(x) are smooth, repeating waves. For y = A sin(Bx), the <strong>amplitude</strong> is |A| (how far the wave reaches above and below the midline), and the <strong>period</strong> (the length of one complete wave) is <span class="frac"><span class="num">2π</span><span class="den">|B|</span></span>.</p>

<p><strong>Worked Example:</strong> For y = 3 sin(x), what is the amplitude?</p>

<p class="step-math">amplitude = |3| = 3</p>

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<p class="prompt">1. For y = 4 cos(x), what is the amplitude?</p>

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<p><span class="label">Explanation:</span> The amplitude is |4| = 4.</p>

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<p class="prompt">2. What is the period of y = sin(x)?</p>

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<p><span class="label">Explanation:</span> The standard sine function completes one full wave every 2π.</p>

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<p class="prompt">3. For y = sin(2x), what is the period?</p>

</div>

<p><span class="label">Explanation:</span> period = <span class="frac"><span class="num">2π</span><span class="den">|B|</span></span> = <span class="frac"><span class="num">2π</span><span class="den">2</span></span> = π.</p>

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<p class="prompt">4. For y = 5 sin(x), what are the maximum and minimum values of y?</p>

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<p><span class="label">Explanation:</span> The amplitude of 5 means the graph reaches 5 units above and below its midline (0), giving a maximum of 5 and a minimum of −5.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Mixing up which side is opposite versus adjacent.</strong> These labels are always relative to the specific angle in question, not fixed positions in the triangle.</li>

<li><strong>Using SOH-CAH-TOA on a non-right triangle.</strong> The basic trig ratios only work in right triangles. For any other triangle, switch to the Law of Sines or Law of Cosines.</li>

<li><strong>Forgetting the minus sign in the Law of Cosines.</strong> It's a² = b² + c² − 2bc·cos A, not plus, this is easy to misremember since it resembles the Pythagorean theorem.</li>

<li><strong>Leaving the calculator in radian mode for a degree-based problem, or vice versa.</strong> Always confirm your calculator's angle mode matches the units given in the problem before computing.</li>

<li><strong>Confusing amplitude with period.</strong> Amplitude describes the wave's height (vertical stretch); period describes how long it takes to repeat (horizontal length). They come from different parts of the equation.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>Is it really true that ACT trigonometry goes beyond SOH-CAH-TOA?</h3>

<p>Yes. While right-triangle ratios cover roughly half of ACT trig questions, a well-rounded prep plan should also include the Law of Sines and Cosines, radian measure and the unit circle, and the basic shape and parameters of sine and cosine graphs.</p>

</div>

<h3>When should I use the Law of Sines versus the Law of Cosines?</h3>

<p>Use the Law of Sines when you know an angle and its opposite side, plus one more piece of information (another angle, or another side-angle pair). Use the Law of Cosines when you know two sides and the included angle, or all three sides, situations where the Law of Sines doesn't have enough matching angle-side pairs to work with.</p>

</div>

<h3>Do I need to memorize every value on the unit circle?</h3>

<p>Knowing the special angles, 0°, 30°, 45°, 60°, and 90°, along with which quadrant an angle falls in, covers the large majority of what's tested. Full unit circle memorization for every angle is not necessary.</p>

</div>

<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-trigonometry-quiz-1">Trigonometry quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-trigonometry-quiz-1">Practice Trigonometry Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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