Question

Solve each equation algebraically over the domain \(0 \leq x<2 \pi\). a) \(\tan ^{2} x-\tan x=0\) b) \(\sin 2 x-\sin x=0\) c) \(\sin ^{2} x-4 \sin x=5\) d) \(\cos 2 x=\sin x\)

Short answer

a) \(x = 0, \frac{\pi}{4}, \pi, \frac{5\pi}{4}\); b) \(x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3}\); c) \(x = \frac{3\pi}{2}\); d) \(x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}\)

Step-by-step solution

Solve Equation a) \( \tan ^{2} x - \tan x = 0 \)

Factor out \( \tan x \) from the left side of the equation: \( \tan x (\tan x - 1) = 0 \). This gives us two solutions: \( \tan x = 0 \) and \( \tan x - 1 = 0 \). Solving these, we get \( \tan x = 0 \) and \( \tan x = 1 \).

Find solutions for \( \tan x = 0 \)

\( \tan x = 0 \) when \( x = 0 \) or \( x = \pi \) within the given domain \( 0 \leq x < 2\pi \).

Find solutions for \( \tan x = 1 \)

\( \tan x = 1 \) when \( x = \frac{\pi}{4} \) or \( x = \frac{5\pi}{4} \) within the given domain \( 0 \leq x < 2\pi \). Thus, solutions for part a) are \( x = 0, \frac{\pi}{4}, \pi, \frac{5\pi}{4} \).

Solve Equation b) \( \sin 2 x - \sin x = 0 \)

Rewrite \( \sin 2x \) as \( 2 \sin x \cos x \): \( 2 \sin x \cos x - \sin x = 0 \). Factor out \( \sin x \): \( \sin x (2 \cos x - 1) = 0 \). This gives us two solutions: \( \sin x = 0 \) and \( 2 \cos x - 1 = 0 \).

Find solutions for \( \sin x = 0 \)

\( \sin x = 0 \) when \( x = 0, \pi, 2\pi \). Since \( x < 2\pi \), the valid solutions within the domain are \( x = 0 \) and \( x = \pi \).

Find solutions for \( 2 \cos x - 1 = 0 \)

\( 2 \cos x - 1 = 0 \) leads to \( \cos x = \frac{1}{2} \) when \( x = \frac{\pi}{3} \) or \( x = \frac{5 \pi}{3} \). Thus, solutions for part b) are \( x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3} \).

Solve Equation c) \( \sin ^{2} x - 4 \sin x = 5 \)

Let \( y = \sin x \). The equation becomes \( y^2 - 4y - 5 = 0 \). Factorize it: \( (y - 5)(y + 1) = 0 \). This gives \( y = 5 \) and \( y = -1 \).

Solve for \( \sin x = 5 \) and \( \sin x = -1 \)

\( \sin x = 5 \) has no solutions as \( \sin x \) must be in \([-1, 1]\). \( \sin x = -1 \) when \( x = \frac{3\pi}{2} \). Thus, the solution for part c) is \( x = \frac{3 \pi}{2} \).

Solve Equation d) \( \cos 2 x = \sin x \)

Rewrite \( \cos 2x \) using its identity: \( \cos 2x = 1 - 2 \sin^2 x \). The equation becomes \( 1 - 2 \sin^2 x = \sin x \). Rearrange it to form a quadratic: \( 2 \sin^2 x + \sin x - 1 = 0 \).

Solve quadratic equation \( 2 \sin^2 x + \sin x - 1 = 0 \)

Use the quadratic formula \( y = \sin x \): \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 2 \), \( b = 1 \), and \( c = -1 \). \( y = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4} \). So, \( y = \frac{1}{2} \) and \( y = -1 \).

Find solutions for \( \sin x = \frac{1}{2} \) and \( \sin x = -1 \)

\( \sin x = \frac{1}{2} \) when \( x = \frac{\pi}{6} \) or \( x = \frac{5 \pi}{6} \). From earlier, \( \sin x = -1 \) when \( x = \frac{3 \pi}{2} \). Thus, solutions for part d) are \( x = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{3 \pi}{2} \).

Key concepts

Trigonometric Identities

Fatal error: Uncaught Error: Object of class stdClass could not be converted to string in /var/www/html/web/app/themes/studypress-core-theme/template-parts/API/TBS/question/intro-section.php:360 Stack trace: #0 /var/www/html/web/wp/wp-includes/template.php(792): require() #1 /var/www/html/web/wp/wp-includes/template.php(725): load_template('/var/www/html/w...', false, Array) #2 /var/www/html/web/wp/wp-includes/general-template.php(206): locate_template(Array, true, false, Array) #3 /var/www/html/web/app/themes/studypress-core-theme/page-templates/API/TBS/question.php(264): get_template_part('template-parts/...', '', Array) #4 /var/www/html/web/wp/wp-includes/template.php(792): require('/var/www/html/w...') #5 /var/www/html/web/wp/wp-includes/template.php(725): load_template('/var/www/html/w...', false, Array) #6 /var/www/html/web/wp/wp-includes/general-template.php(206): locate_template(Array, true, false, Array) #7 /var/www/html/web/app/themes/studypress-core-theme/inc/Theme/Api_Pages/Abstract_Api_Page.php(70): get_template_part('page-templates/...', '', Array) #8 /var/www/html/web/wp/wp-includes/class-wp-hook.php(324): StudySmarter\Theme\Api_Pages\Abstract_Api_Page->template_redirect('') #9 /var/www/html/web/wp/wp-includes/class-wp-hook.php(348): WP_Hook->apply_filters(NULL, Array) #10 /var/www/html/web/wp/wp-includes/plugin.php(517): WP_Hook->do_action(Array) #11 /var/www/html/web/wp/wp-includes/template-loader.php(13): do_action('template_redire...') #12 /var/www/html/web/wp/wp-blog-header.php(19): require_once('/var/www/html/w...') #13 /var/www/html/web/index.php(6): require('/var/www/html/w...') #14 {main} thrown in /var/www/html/web/app/themes/studypress-core-theme/template-parts/API/TBS/question/intro-section.php on line 360