{"id":140,"date":"2025-03-04T19:55:21","date_gmt":"2025-03-04T19:55:21","guid":{"rendered":"https:\/\/sebstack.com\/?page_id=140"},"modified":"2026-01-03T21:38:37","modified_gmt":"2026-01-03T21:38:37","slug":"math-tips-and-tricks","status":"publish","type":"page","link":"https:\/\/sebstack.com\/index.php\/math-tips-and-tricks\/","title":{"rendered":"Math Tips and Tricks"},"content":{"rendered":"\n<p>WildTrig playlist &#8211; <a href=\"https:\/\/youtube.com\/playlist?list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;si=yN_j1mo9EysTcfoq\">https:\/\/youtube.com\/playlist?list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;si=yN_j1mo9EysTcfoq<\/a><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>1 Why Trig is Hard | WildTrig: Intro to Rational Trigonometry | N J Wildberger<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Why Trig is Hard | WildTrig: Intro to Rational Trigonometry | N J Wildberger\" width=\"625\" height=\"469\" src=\"https:\/\/www.youtube.com\/embed\/ZYWHfvij94U?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ZYWHfvij94U&amp;list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;index=1\">00:00<\/a> Introduction to the series on rational trig <a href=\"https:\/\/www.youtube.com\/watch?v=ZYWHfvij94U&amp;list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;index=1&amp;t=60s\">01:00<\/a> Trigonometry and triangle measurements <a href=\"https:\/\/www.youtube.com\/watch?v=ZYWHfvij94U&amp;list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;index=1&amp;t=194s\">03:14<\/a> Distance and angle: are they really fundamental? <a href=\"https:\/\/www.youtube.com\/watch?v=ZYWHfvij94U&amp;list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;index=1&amp;t=373s\">06:13<\/a> The classical sine function and intrinsic irrationalities<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"535\" height=\"258\" src=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image.png\" alt=\"\" class=\"wp-image-375\" srcset=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image.png 535w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-300x145.png 300w\" sizes=\"auto, (max-width: 535px) 100vw, 535px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"721\" height=\"224\" src=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-1.png\" alt=\"\" class=\"wp-image-377\" srcset=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-1.png 721w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-1-300x93.png 300w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-1-624x194.png 624w\" sizes=\"auto, (max-width: 721px) 100vw, 721px\" \/><\/figure>\n\n\n\n<p>2.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Quadrance via Pythagoras and Archimedes | WildTrig: Intro to Rational Trigonometry | N J Wildberger\" width=\"625\" height=\"469\" src=\"https:\/\/www.youtube.com\/embed\/3GU9mGyxz04?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"891\" height=\"490\" src=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-2.png\" alt=\"\" class=\"wp-image-379\" srcset=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-2.png 891w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-2-300x165.png 300w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-2-768x422.png 768w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-2-624x343.png 624w\" sizes=\"auto, (max-width: 891px) 100vw, 891px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"616\" height=\"508\" src=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-3.png\" alt=\"\" class=\"wp-image-380\" srcset=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-3.png 616w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-3-300x247.png 300w\" sizes=\"auto, (max-width: 616px) 100vw, 616px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"748\" height=\"433\" src=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-4.png\" alt=\"\" class=\"wp-image-383\" srcset=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-4.png 748w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-4-300x174.png 300w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/image-4-624x361.png 624w\" sizes=\"auto, (max-width: 748px) 100vw, 748px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p>3.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Spread, Angles and Astronomy | WildTrig: Intro to Rational Trigonometry | N J Wildberger\" width=\"625\" height=\"469\" src=\"https:\/\/www.youtube.com\/embed\/9wd0i44vK04?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"568\" height=\"319\" src=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/A-spread-protractor.png\" alt=\"\" class=\"wp-image-386\" style=\"aspect-ratio:1.7805717198317972;width:655px;height:auto\" srcset=\"https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/A-spread-protractor.png 568w, https:\/\/sebstack.com\/wp-content\/uploads\/2026\/01\/A-spread-protractor-300x168.png 300w\" sizes=\"auto, (max-width: 568px) 100vw, 568px\" \/><\/figure>\n<\/div>\n\n\n<p>source: <a href=\"https:\/\/www.researchgate.net\/figure\/A-spread-protractor_fig4_284572258\">https:\/\/www.researchgate.net\/figure\/A-spread-protractor_fig4_284572258<\/a><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>4.  Five Main Laws of Rational Trigonometry | WildTrig: Intro to Rational Trigonometry | N J Wildberger<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Five Main Laws of Rational Trigonometry | WildTrig: Intro to Rational Trigonometry | N J Wildberger\" width=\"625\" height=\"469\" src=\"https:\/\/www.youtube.com\/embed\/Oe2DZc6BXZk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p>27. Stewart&#8217;s theorem | WildTrig:  N J Wildberger <\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Stewart&#039;s theorem | WildTrig: Intro to Rational Trigonometry | N J Wildberger\" width=\"625\" height=\"469\" src=\"https:\/\/www.youtube.com\/embed\/cRebl8I0lKk?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p>Trigonometric identities play a crucial role in simplifying equations in mathematics, physics, and engineering. Below is a list of fundamental trigonometric identities, carefully formatted to work on your website.<\/p>\n\n\n\n<p>Find in page how Pythagorean Identities are used here: <\/p>\n\n\n\n<p><a href=\"#Peak-and-RMS-voltage\" data-type=\"internal\" data-id=\"#Peak-and-RMS-voltage\">Peak and RMS voltage<\/a><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. Pythagorean Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin^2 x + \\cos^2 x = 1 [\/math]<\/li>\n\n\n\n<li>[math] 1 + \\tan^2 x = \\sec^2 x [\/math]<\/li>\n\n\n\n<li>[math] 1 + \\cot^2 x = \\csc^2 x [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Reciprocal Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin x = \\frac{1}{\\csc x} [\/math], [math] \\csc x = \\frac{1}{\\sin x} [\/math]<\/li>\n\n\n\n<li>[math] \\cos x = \\frac{1}{\\sec x} [\/math], [math] \\sec x = \\frac{1}{\\cos x} [\/math]<\/li>\n\n\n\n<li>[math] \\tan x = \\frac{1}{\\cot x} [\/math], [math] \\cot x = \\frac{1}{\\tan x} [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>3. Quotient Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\tan x = \\frac{\\sin x}{\\cos x} [\/math]<\/li>\n\n\n\n<li>[math] \\cot x = \\frac{\\cos x}{\\sin x} [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>4. Co-Function Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin(90^\\circ &#8211; x) = \\cos x [\/math], [math] \\cos(90^\\circ &#8211; x) = \\sin x [\/math]<\/li>\n\n\n\n<li>[math] \\tan(90^\\circ &#8211; x) = \\cot x [\/math], [math] \\cot(90^\\circ &#8211; x) = \\tan x [\/math]<\/li>\n\n\n\n<li>[math] \\sec(90^\\circ &#8211; x) = \\csc x [\/math], [math] \\csc(90^\\circ &#8211; x) = \\sec x [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>5. Even-Odd Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin(-x) = -\\sin x [\/math]<\/li>\n\n\n\n<li>[math] \\cos(-x) = \\cos x [\/math]<\/li>\n\n\n\n<li>[math] \\tan(-x) = -\\tan x [\/math]<\/li>\n\n\n\n<li>[math] \\cot(-x) = -\\cot x [\/math]<\/li>\n\n\n\n<li>[math] \\sec(-x) = \\sec x [\/math]<\/li>\n\n\n\n<li>[math] \\csc(-x) = -\\csc x [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>6. Sum and Difference Formulas<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B [\/math]<\/li>\n\n\n\n<li>[math] \\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B [\/math]<\/li>\n\n\n\n<li>[math] \\tan(A \\pm B) = \\frac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B} [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>7. Double-Angle Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin 2x = 2 \\sin x \\cos x [\/math]<\/li>\n\n\n\n<li>[math] \\cos 2x = \\cos^2 x &#8211; \\sin^2 x [\/math]<\/li>\n\n\n\n<li>Alternative forms:\n<ul class=\"wp-block-list\">\n<li>[math] \\cos 2x = 2 \\cos^2 x &#8211; 1 [\/math]<\/li>\n\n\n\n<li>[math] \\cos 2x = 1 &#8211; 2 \\sin^2 x [\/math]<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>[math] \\tan 2x = \\frac{2 \\tan x}{1 &#8211; \\tan^2 x} [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>8. Half-Angle Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin \\frac{x}{2} = \\pm \\sqrt{\\frac{1 &#8211; \\cos x}{2}} [\/math]<\/li>\n\n\n\n<li>[math] \\cos \\frac{x}{2} = \\pm \\sqrt{\\frac{1 + \\cos x}{2}} [\/math]<\/li>\n\n\n\n<li>[math] \\tan \\frac{x}{2} = \\pm \\sqrt{\\frac{1 &#8211; \\cos x}{1 + \\cos x}} [\/math]<\/li>\n\n\n\n<li>Alternative forms:\n<ul class=\"wp-block-list\">\n<li>[math] \\tan \\frac{x}{2} = \\frac{\\sin x}{1 + \\cos x} [\/math]<\/li>\n\n\n\n<li>[math] \\tan \\frac{x}{2} = \\frac{1 &#8211; \\cos x}{\\sin x} [\/math]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>9. Product-to-Sum Identities<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin A \\sin B = \\frac{1}{2} [\\cos(A &#8211; B) &#8211; \\cos(A + B)] [\/math]<\/li>\n\n\n\n<li>[math] \\cos A \\cos B = \\frac{1}{2} [\\cos(A &#8211; B) + \\cos(A + B)] [\/math]<\/li>\n\n\n\n<li>[math] \\sin A \\cos B = \\frac{1}{2} [\\sin(A + B) + \\sin(A &#8211; B)] [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<h3 class=\"wp-block-heading\"><strong>10. Sum-to-Product Identities<\/strong><\/h3>\n<\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\sin A + \\sin B = 2 \\sin \\frac{A + B}{2} \\cos \\frac{A &#8211; B}{2} [\/math]<\/li>\n\n\n\n<li>[math] \\sin A &#8211; \\sin B = 2 \\cos \\frac{A + B}{2} \\sin \\frac{A &#8211; B}{2} [\/math]<\/li>\n\n\n\n<li>[math] \\cos A + \\cos B = 2 \\cos \\frac{A + B}{2} \\cos \\frac{A &#8211; B}{2} [\/math]<\/li>\n\n\n\n<li>[math] \\cos A &#8211; \\cos B = -2 \\sin \\frac{A + B}{2} \\sin \\frac{A &#8211; B}{2} [\/math]<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>11. Special Trigonometric Identities for AC Circuits<\/strong><\/h3>\n\n\n\n<p>These identities are particularly useful in <strong>Peak and RMS Voltage<\/strong> calculations.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[math] \\cos^2 x = \\frac{1 + \\cos 2x}{2} [\/math]<\/li>\n\n\n\n<li>[math] \\sin^2 x = \\frac{1 &#8211; \\cos 2x}{2} [\/math]<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>3. Quotient Identities<\/strong><\/h3>\n\n\n\n<p><\/p>\n\n\n\n<p>Mathematics is full of elegant identities that simplify complex problems. One of the most useful identities in electrical engineering appears when calculating <strong>instantaneous power in AC circuits<\/strong>. This identity is:<\/p>\n\n\n\n<p>[math] \\cos^2 x = \\frac{1 + \\cos 2x}{2} [\/math]<\/p>\n\n\n\n<p>This trigonometric property naturally emerges when dealing with <strong>Peak and RMS Voltage<\/strong>, helping us understand how power oscillates in an AC circuit.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"Peak-and-RMS-voltage\"><strong>Peak and RMS Voltage Calculation Using Trigonometric Identities<\/strong><\/h3>\n\n\n\n<p>In an AC circuit with a purely resistive load, the voltage and current follow sinusoidal waveforms:<\/p>\n\n\n\n<p>[math] v(t) = V_{\\text{peak}} \\cos(\\omega t) [\/math]<\/p>\n\n\n\n<p>[math] i(t) = I_{\\text{peak}} \\cos(\\omega t) [\/math]<\/p>\n\n\n\n<p>The <strong>instantaneous power<\/strong> is given by:<\/p>\n\n\n\n<p>[math] P(t) = v(t) \\cdot i(t) [\/math]<\/p>\n\n\n\n<p>Substituting the voltage and current expressions:<\/p>\n\n\n\n<p>[math] P(t) = V_{\\text{peak}} \\cos(\\omega t) \\cdot I_{\\text{peak}} \\cos(\\omega t) [\/math]<\/p>\n\n\n\n<p>Using the <strong>trigonometric identity<\/strong>:<\/p>\n\n\n\n<p>[math] \\cos^2 x = \\frac{1 + \\cos(2x)}{2} [\/math]<\/p>\n\n\n\n<p>we can rewrite the power equation as:<\/p>\n\n\n\n<p>[math] P(t) = V_{\\text{peak}} I_{\\text{peak}} \\frac{1 + \\cos(2\\omega t)}{2} [\/math]<\/p>\n\n\n\n<p>or:<\/p>\n\n\n\n<p>[math] P(t) = \\frac{V_{\\text{peak}} I_{\\text{peak}}}{2} (1 + \\cos(2\\omega t)) [\/math]<\/p>\n\n\n\n<p>This shows that the power oscillates at <strong>twice the frequency<\/strong> of the original voltage and current waveforms.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Converting to RMS Voltage<\/strong><\/h3>\n\n\n\n<p>The <strong>RMS (Root Mean Square) voltage<\/strong> is related to the peak voltage by:<\/p>\n\n\n\n<p>[math] V_{\\text{rms}} = \\frac{V_{\\text{peak}}}{\\sqrt{2}} [\/math]<\/p>\n\n\n\n<p>Squaring both sides:<\/p>\n\n\n\n<p>[math] V_{\\text{rms}}^2 = \\frac{V_{\\text{peak}}^2}{2} [\/math]<\/p>\n\n\n\n<p>This allows us to rewrite the power equation in terms of <strong>RMS voltage<\/strong>:<\/p>\n\n\n\n<p>[math] P(t) = \\frac{V_{\\text{rms}}^2}{R} (1 + \\cos(2\\omega t)) [\/math]<\/p>\n\n\n\n<p>Here, the <strong>factor of 12\\frac{1}{2}21\u200b disappears<\/strong>, because it has already been incorporated into the RMS definition.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conclusion<\/strong><\/h3>\n\n\n\n<p>The trigonometric identity:<\/p>\n\n\n\n<p>[math] \\cos^2 x = \\frac{1 + \\cos 2x}{2} [\/math]<\/p>\n\n\n\n<p>naturally appears in <strong>AC power calculations<\/strong>, simplifying the expression for <strong>instantaneous power<\/strong> and justifying the use of <strong>RMS values<\/strong> in electrical engineering.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Math Tips and Tricks: Trigonometric Identities - When They Appear\n\nIn electrical engineering, particularly when dealing with alternating current (AC) signals, we frequently encounter the need to calculate peak and root-mean-square (RMS) voltages. These calculations often involve trigonometric identities, specifically those related to the cosine function. Let's delve into how the identity leading to (1 + cos(2\u03c9t)) arises and its significance.\n\nUnderstanding the Foundation\n\nConsider a sinusoidal AC voltage signal, represented by:<\/code><\/pre>\n\n\n<p>[math]v(t) = V_p \\cos(\\omega t)[\/math]<\/p>\n\n\n\n<p>where: * v(t) is the instantaneous voltage at time t. * Vp is the peak voltage. * \u03c9 is the angular frequency. To calculate the RMS voltage, we need to find the square root of the mean of the squared voltage over one period. This involves squaring the voltage signal:<\/p>\n\n\n<p>[math]v^2(t) = (V_p \\cos(\\omega t))^2 = V_p^2 \\cos^2(\\omega t)[\/math]<\/p>\n\n\n\n<p>Here&#8217;s where the trigonometric identity comes into play. We need to express cos\u00b2(\u03c9t) in a form that&#8217;s easier to integrate over a period. We use the double-angle identity:<\/p>\n\n\n<p>[math]\\cos(2\\theta) = 2\\cos^2(\\theta) &#8211; 1[\/math]<\/p>\n\n\n\n<p>Rearranging this identity to solve for cos\u00b2(\u03b8), we get:<\/p>\n\n\n<p>[math]\\cos^2(\\theta) = \\frac{1 + \\cos(2\\theta)}{2}[\/math]<\/p>\n\n\n\n<p>Substituting \u03c9t for \u03b8, we have:<\/p>\n\n\n<p>[math]\\cos^2(\\omega t) = \\frac{1 + \\cos(2\\omega t)}{2}[\/math]<\/p>\n\n\n\n<p>Therefore, the squared voltage becomes:<\/p>\n\n\n<p>[math]v^2(t) = V_p^2 \\left(\\frac{1 + \\cos(2\\omega t)}{2}\\right)[\/math]<\/p>\n\n\n\n<p>RMS Voltage Calculation The RMS voltage (Vrms) is calculated as:<\/p>\n\n\n<p>[math]V_{rms} = \\sqrt{\\frac{1}{T} \\int_0^T v^2(t) dt}[\/math]<\/p>\n\n\n\n<p>where T is the period of the signal. Substituting the expression for v\u00b2(t):<\/p>\n\n\n<p>[math]V_{rms} = \\sqrt{\\frac{1}{T} \\int_0^T V_p^2 \\left(\\frac{1 + \\cos(2\\omega t)}{2}\\right) dt}[\/math]<\/p>\n\n\n\n<p>[math]V_{rms} = V_p \\sqrt{\\frac{1}{2T} \\int_0^T (1 + \\cos(2\\omega t)) dt}[\/math] The integral of cos(2\u03c9t) over one period is zero, leaving:<\/p>\n\n\n<p>[math]V_{rms} = V_p \\sqrt{\\frac{1}{2T} \\int_0^T 1 dt} = V_p \\sqrt{\\frac{1}{2T} \\cdot T} = \\frac{V_p}{\\sqrt{2}}[\/math]<\/p>\n\n\n\n<p>Significance of the Identity The trigonometric identity cos\u00b2(\u03c9t) = (1 + cos(2\u03c9t))\/2 is crucial because it transforms a squared trigonometric function into a form that&#8217;s easily integrated. This allows us to derive the relationship between peak and RMS voltages, which is fundamental in AC circuit analysis. In summary, the appearance of the term (1 + cos(2\u03c9t)) stems directly from the double-angle identity for cosine, facilitating the calculation of RMS voltage from peak voltage.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>WildTrig playlist &#8211; https:\/\/youtube.com\/playlist?list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&amp;si=yN_j1mo9EysTcfoq 1 Why Trig is Hard | WildTrig: Intro to Rational Trigonometry | N J Wildberger 00:00 Introduction to the series on rational trig 01:00 Trigonometry and triangle measurements 03:14 Distance and angle: are they really fundamental? 06:13 The classical sine function and intrinsic irrationalities 2. 3. source: https:\/\/www.researchgate.net\/figure\/A-spread-protractor_fig4_284572258 4. Five Main [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-140","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/pages\/140","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/comments?post=140"}],"version-history":[{"count":16,"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/pages\/140\/revisions"}],"predecessor-version":[{"id":387,"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/pages\/140\/revisions\/387"}],"wp:attachment":[{"href":"https:\/\/sebstack.com\/index.php\/wp-json\/wp\/v2\/media?parent=140"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}