Solve cos ⁡ 2 x − cos ⁡ x = 0 \cos2x-\cos x=0 cos 2 x − cos x = 0 for 0 ≤ x ≤ 2 π 0\le x\le2\pi 0 ≤ x ≤ 2 π .

0 , 2 π 3 , 4 π 3 , 2 π 0,\ \frac{2\pi}{3},\ \frac{4\pi}{3},\ 2\pi 0 , 3 2 π , 3 4 π , 2 π

5 π 6 , 7 π 6 \frac{5\pi}{6},\ \frac{7\pi}{6} 6 5 π , 6 7 π

π 3 , 5 π 3 \frac{\pi}{3},\ \frac{5\pi}{3} 3 π , 3 5 π

π 4 , 3 π 4 , 5 π 4 \frac{\pi}{4},\frac{3\pi}{4},\ \frac{5\pi}{4} 4 π , 4 3 π , 4 5 π

Given that sin ⁡ θ = − 5 6 , 3 π 2 ≤ θ ≤ 2 π \sin\theta=-\frac{5}{6},\ \frac{3\pi}{2}\le\theta\le2\pi sin θ = − 6 5 , 2 3 π ≤ θ ≤ 2 π , find cos ⁡ 2 θ \cos\ 2\theta cos 2 θ .

− 7 18 -\frac{7}{18} − 1 8 7

− 5 11 18 -\frac{5\sqrt{11}}{18} − 1 8 5 1 1 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

− 10 12 -\frac{10}{12} − 1 2 1 0

If cos ⁡ β = 2 5 \cos\beta=\frac{2}{5} cos β = 5 2 where 270 ° ≤ β ≤ 360 ° 270\degree\le\beta\le360\degree 2 7 0 ° ≤ β ≤ 3 6 0 ° , find sin ⁡ 2 β \sin2\beta sin 2 β .

− 4 21 25 -\frac{4\sqrt{21}}{25} − 2 5 4 2 1 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

4 21 25 \frac{4\sqrt{21}}{25} 2 5 4 2 1 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

4 3 5 \frac{4\sqrt{3}}{5} 5 4 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

− 4 3 5 -\frac{4\sqrt{3}}{5} − 5 4 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Solve for 0 ≤ x ≤ 2 π 0\le x\le2\pi 0 ≤ x ≤ 2 π , sin ⁡ x = sin ⁡ 2 x \sin x=\sin\ 2x sin x = sin 2 x .

0 , π 3 , π , 5 π 3 , 2 π 0,\ \frac{\pi}{3},\ \pi,\ \frac{5\pi}{3},\ 2\pi 0 , 3 π , π , 3 5 π , 2 π

0 , π 6 , 11 π 6 , 2 π 0,\ \frac{\pi}{6},\frac{11\pi}{6},\ 2\pi 0 , 6 π , 6 1 1 π , 2 π

Solve sin ⁡ θ cos ⁡ θ = 1 4 \sin\theta\cos\theta=\frac{1}{4} sin θ cos θ = 4 1 for − π ≤ θ ≤ π -\pi\le\theta\le\pi − π ≤ θ ≤ π .

− 11 π 12 , − 7 π 12 , π 12 , 5 π 12 -\frac{11\pi}{12},-\frac{7\pi}{12},\frac{\pi}{12},\frac{5\pi}{12} − 1 2 1 1 π , − 1 2 7 π , 1 2 π , 1 2 5 π

± π 12 , ± 5 π 12 \pm\frac{\pi}{12},\pm\frac{5\pi}{12} ± 1 2 π , ± 1 2 5 π

− 7 π 6 , − 11 π 6 , π 6 , 5 π 6 -\frac{7\pi}{6},-\frac{11\pi}{6},\frac{\pi}{6},\ \frac{5\pi}{6} − 6 7 π , − 6 1 1 π , 6 π , 6 5 π

± π 6 , ± 5 π 6 \pm\frac{\pi}{6},\pm\frac{5\pi}{6} ± 6 π , ± 6 5 π

If sin ⁡ θ = 4 5 \sin\theta=\frac{4}{5} sin θ = 5 4 and cos ⁡ θ = 3 5 \cos\theta=\frac{3}{5} cos θ = 5 3 find tan ⁡ 2 θ \tan2\theta tan 2 θ .

− 24 7 -\frac{24}{7} − 7 2 4

− 7 24 -\frac{7}{24} − 2 4 7

Trigonometry - Identities and double angle formulas

Authored by Karen Carcamo

Other

10th - 12th Grade

CCSS covered

Used 71+ times

Trigonometry - Identities and double angle formulas

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10 questions

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MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Solve $\cos2x-\cos x=0$ for $0\le x\le2\pi$ .

$0,\ \frac{2\pi}{3},\ \frac{4\pi}{3},\ 2\pi$

$\frac{5\pi}{6},\ \frac{7\pi}{6}$

$\frac{\pi}{3},\ \frac{5\pi}{3}$

$\frac{\pi}{4},\frac{3\pi}{4},\ \frac{5\pi}{4}$

Tags

CCSS.HSF.TF.B.7

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Given that $\sin\theta=-\frac{5}{6},\ \frac{3\pi}{2}\le\theta\le2\pi$ , find $\cos\ 2\theta$ .

$\frac{10}{12}$

$-\frac{5\sqrt{11}}{18}$

$-\frac{7}{18}$

$-\frac{10}{12}$

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

If $\cos\beta=\frac{2}{5}$ where $270\degree\le\beta\le360\degree$ , find $\sin2\beta$ .

$-\frac{4\sqrt{21}}{25}$

$\frac{4\sqrt{21}}{25}$

$\frac{4\sqrt{3}}{5}$

$-\frac{4\sqrt{3}}{5}$

Tags

CCSS.HSF.TF.C.9

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Simplify $\frac{\sin\left(\pi-\theta\right)-\sin\left(-\theta\right)}{\cos\ \left(-\theta\right)}$

$0$

$2\tan\theta$

$\frac{1}{\cos\ \theta}$

$\tan\theta$

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Solve for $0\le x\le2\pi$ , $\sin x=\sin\ 2x$ .

$0,\ \frac{\pi}{3},\ \pi,\ \frac{5\pi}{3},\ 2\pi$

$0,\ 2\pi$

$0,\ \frac{\pi}{6},\frac{11\pi}{6},\ 2\pi$

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Solve $\sin\theta\cos\theta=\frac{1}{4}$ for $-\pi\le\theta\le\pi$ .

$\pm\frac{\pi}{12},\pm\frac{5\pi}{12}$

$-\frac{7\pi}{6},-\frac{11\pi}{6},\frac{\pi}{6},\ \frac{5\pi}{6}$

$\pm\frac{\pi}{6},\pm\frac{5\pi}{6}$

$-\frac{11\pi}{12},-\frac{7\pi}{12},\frac{\pi}{12},\frac{5\pi}{12}$

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

If $\sin\theta=\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$ find $\tan2\theta$ .

$-\frac{7}{24}$

$\frac{24}{7}$

$\frac{7}{24}$

$-\frac{24}{7}$

Tags

CCSS.HSF.TF.C.9

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Simplify $\left(2\sin\theta+3\cos\theta\right)^2+\left(3\sin\theta-2\cos\theta\right)^2$ .

$5$

$13\sin^2\ \theta\ +\ 24\sin\theta\cos\theta+13\cos^2\ \theta$

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

If $\sin\alpha=-\frac{2}{3}$ , find $\cos2\alpha$ .

$\frac{5}{9}$

$\frac{7}{3}$

$\frac{1}{9}$

$\frac{8}{9}$

Tags

CCSS.HSF.TF.C.9

10.

MULTIPLE CHOICE QUESTION

15 mins • 1 pt

Find the exact answer of $\left(\cos\ \frac{\pi}{12}+\sin\ \frac{\pi}{12}\right)^2$ .

$-\frac{1}{2}$

$1$

$\frac{1}{2}$

$\frac{3}{2}$

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Trigonometry - Identities and double angle formulas

Solve cos⁡2x−cos⁡x=0\cos2x-\cos x=0cos2x−cosx=0 for 0≤x≤2π0\le x\le2\pi0≤x≤2π .

Given that sin⁡θ=−56, 3π2≤θ≤2π\sin\theta=-\frac{5}{6},\ \frac{3\pi}{2}\le\theta\le2\pisinθ=−65​, 23π​≤θ≤2π , find cos⁡ 2θ\cos\ 2\thetacos 2θ .

If cos⁡β=25\cos\beta=\frac{2}{5}cosβ=52​ where 270°≤β≤360°270\degree\le\beta\le360\degree270°≤β≤360° , find sin⁡2β\sin2\betasin2β .

Simplify sin⁡(π−θ)−sin⁡(−θ)cos⁡ (−θ)\frac{\sin\left(\pi-\theta\right)-\sin\left(-\theta\right)}{\cos\ \left(-\theta\right)}cos (−θ)sin(π−θ)−sin(−θ)​

Solve for 0≤x≤2π0\le x\le2\pi0≤x≤2π , sin⁡x=sin⁡ 2x\sin x=\sin\ 2xsinx=sin 2x .

Solve sin⁡θcos⁡θ=14\sin\theta\cos\theta=\frac{1}{4}sinθcosθ=41​ for −π≤θ≤π-\pi\le\theta\le\pi−π≤θ≤π .

If sin⁡θ=45\sin\theta=\frac{4}{5}sinθ=54​ and cos⁡θ=35\cos\theta=\frac{3}{5}cosθ=53​ find tan⁡2θ\tan2\thetatan2θ .

Simplify (2sin⁡θ+3cos⁡θ)2+(3sin⁡θ−2cos⁡θ)2\left(2\sin\theta+3\cos\theta\right)^2+\left(3\sin\theta-2\cos\theta\right)^2(2sinθ+3cosθ)2+(3sinθ−2cosθ)2 .

If sin⁡α=−23\sin\alpha=-\frac{2}{3}sinα=−32​ , find cos⁡2α\cos2\alphacos2α .

Find the exact answer of (cos⁡ π12+sin⁡ π12)2\left(\cos\ \frac{\pi}{12}+\sin\ \frac{\pi}{12}\right)^2(cos 12π​+sin 12π​)2 .

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Solve $\cos2x-\cos x=0$ for $0\le x\le2\pi$ .

Given that $\sin\theta=-\frac{5}{6},\ \frac{3\pi}{2}\le\theta\le2\pi$ , find $\cos\ 2\theta$ .

If $\cos\beta=\frac{2}{5}$ where $270\degree\le\beta\le360\degree$ , find $\sin2\beta$ .

Simplify $\frac{\sin\left(\pi-\theta\right)-\sin\left(-\theta\right)}{\cos\ \left(-\theta\right)}$

Solve for $0\le x\le2\pi$ , $\sin x=\sin\ 2x$ .

Solve $\sin\theta\cos\theta=\frac{1}{4}$ for $-\pi\le\theta\le\pi$ .

If $\sin\theta=\frac{4}{5}$ and $\cos\theta=\frac{3}{5}$ find $\tan2\theta$ .

Simplify $\left(2\sin\theta+3\cos\theta\right)^2+\left(3\sin\theta-2\cos\theta\right)^2$ .

If $\sin\alpha=-\frac{2}{3}$ , find $\cos2\alpha$ .

Find the exact answer of $\left(\cos\ \frac{\pi}{12}+\sin\ \frac{\pi}{12}\right)^2$ .