Calculate the value(s) of θ \theta θ such that cos ⁡ ( θ ) = − 1 2 \cos\left(\theta\right)=-\frac{1}{2} cos ( θ ) = − 2 1 for 0 ≤ θ ≤ 2 π 0\le\theta\le2\pi 0 ≤ θ ≤ 2 π .

θ = 2 π 3 , 4 π 3 \theta=\frac{2\pi}{3},\frac{4\pi}{3} θ = 3 2 π , 3 4 π

θ = π 3 , 2 π 3 \theta=\frac{\pi}{3},\frac{2\pi}{3} θ = 3 π , 3 2 π

θ = 2 π 3 \theta=\frac{2\pi}{3} θ = 3 2 π

θ = π 3 , 5 π 3 \theta=\frac{\pi}{3},\frac{5\pi}{3} θ = 3 π , 3 5 π

Calculate the value(s) of θ \theta θ such that tan ⁡ ( 2 θ ) = 3 \tan\left(2\theta\right)=\sqrt{3} tan ( 2 θ ) = 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"> for 0 ≤ θ ≤ 2 π 0\le\theta\le2\pi 0 ≤ θ ≤ 2 π .

θ = π 6 , 2 π 3 , 7 π 6 , 6 π 5 \theta=\frac{\pi}{6},\frac{2\pi}{3},\frac{\ 7\pi}{6},\frac{6\pi}{5} θ = 6 π , 3 2 π , 6 7 π , 5 6 π

θ = π 6 , 2 π 3 \theta=\frac{\pi}{6},\frac{2\pi}{3} θ = 6 π , 3 2 π

θ = π 3 , 4 π 3 , 7 π 3 , 12 π 5 \theta=\frac{\pi}{3},\frac{4\pi}{3},\frac{\ 7\pi}{3},\frac{12\pi}{5} θ = 3 π , 3 4 π , 3 7 π , 5 1 2 π

θ = 7 π 6 , 6 π 5 \theta=\frac{\ 7\pi}{6},\frac{6\pi}{5} θ = 6 7 π , 5 6 π

Solve the equation 2 sin ⁡ ( θ ) = tan ⁡ ( θ ) 2\sin\left(\theta\right)=\tan\left(\theta\right) 2 sin ( θ ) = tan ( θ ) for 0 ∘ ≤ θ ≤ 36 0 ∘ 0^{\circ}\le\theta\le360^{\circ} 0 ∘ ≤ θ ≤ 3 6 0 ∘ . Note: Use the fact that tan ⁡ ( θ ) = sin ⁡ ( θ ) cos ⁡ ( θ ) \tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)} tan ( θ ) = cos ( θ ) sin ( θ ) .

θ = 0 ∘ , 6 0 ∘ , 18 0 ∘ , 30 0 ∘ , 36 0 ∘ \theta=0^{\circ},60^{\circ},180^{\circ},300^{\circ},360^{\circ} θ = 0 ∘ , 6 0 ∘ , 1 8 0 ∘ , 3 0 0 ∘ , 3 6 0 ∘

θ = 0 ∘ , 3 0 ∘ , 9 0 ∘ , 15 0 ∘ , 18 0 ∘ \theta=0^{\circ},30^{\circ},90^{\circ},150^{\circ},180^{\circ} θ = 0 ∘ , 3 0 ∘ , 9 0 ∘ , 1 5 0 ∘ , 1 8 0 ∘

θ = 0 ∘ , 18 0 ∘ , 36 0 ∘ \theta=0^{\circ},180^{\circ},360^{\circ} θ = 0 ∘ , 1 8 0 ∘ , 3 6 0 ∘

θ = 0 ∘ , 30 0 ∘ \theta=0^{\circ},300^{\circ} θ = 0 ∘ , 3 0 0 ∘

A right-angled triangle XYZ has an angle, \theta θ , where sin ⁡ ( θ ) = 5 5 \sin\left(\theta\right)=\frac{\sqrt{5}}{5} sin ( θ ) = 5 5 <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"> . Without evaluating θ \theta θ , calculate the exact value(in surd form if applicable) of cos ⁡ ( θ ) \cos\left(\theta\right) cos ( θ ) .

cos ⁡ ( θ ) = 2 5 5 \cos\left(\theta\right)=\frac{2\sqrt{5}}{5} cos ( θ ) = 5 2 5 <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">

cos ⁡ ( θ ) = 2 5 \cos\left(\theta\right)=2\sqrt{5} cos ( θ ) = 2 5 <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">

cos ⁡ ( θ ) = 2 5 \cos\left(\theta\right)=\frac{2}{5} cos ( θ ) = 5 2

cos ⁡ ( θ ) = 5 5 \cos\left(\theta\right)=\frac{\sqrt{5}}{5} cos ( θ ) = 5 5 <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">

A right-angled triangle XYZ has an angle, \theta θ , where sin ⁡ ( θ ) = 5 5 \sin\left(\theta\right)=\frac{\sqrt{5}}{5} sin ( θ ) = 5 5 <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"> . Without evaluating θ \theta θ , calculate the exact value(in surd form if applicable) of sin ⁡ ( 2 θ ) \sin\left(2\theta\right) sin ( 2 θ ) . Note: Use the fact that sin ⁡ ( 2 θ ) = 2 sin ⁡ ( θ ) cos ⁡ ( θ ) \sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right) sin ( 2 θ ) = 2 sin ( θ ) cos ( θ ) .

sin ⁡ ( 2 θ ) = 4 5 \sin\left(2\theta\right)=\frac{4}{5} sin ( 2 θ ) = 5 4

sin ⁡ ( 2 θ ) = 1 5 \sin\left(2\theta\right)=\frac{1}{5} sin ( 2 θ ) = 5 1

sin ⁡ ( 2 θ ) = 2 5 5 \sin\left(2\theta\right)=\frac{2\sqrt{5}}{5} sin ( 2 θ ) = 5 2 5 <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">

sin ⁡ ( 2 θ ) = 10 5 \sin\left(2\theta\right)=\frac{\sqrt{10}}{5} sin ( 2 θ ) = 5 1 0 <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">

Trigonometry (Intermediate)

Authored by Mr Tsoi-a-Sue

Mathematics

11th - 12th Grade

Used 6+ times

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10 questions

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MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which of the following graphs represent the function $f\left(x\right)=\cos\left(2x\right)$ ?

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Which of the following is true?

Sine is positive in the first and third quadrants.

Cosine is negative in the first and fourth quadrants.

Tangent is negative in the second and fourth quadrants.

Sine is negative in the second and third quadrants.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Calculate the value(s) of $\theta$ such that $\cos\left(\theta\right)=-\frac{1}{2}$ for $0\le\theta\le2\pi$ .

$\theta=\frac{2\pi}{3},\frac{4\pi}{3}$

$\theta=\frac{\pi}{3},\frac{2\pi}{3}$

$\theta=\frac{2\pi}{3}$

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

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Calculate the value(s) of $\theta$ such that $\tan\left(2\theta\right)=\sqrt{3}$ for $0\le\theta\le2\pi$ .

$\theta=\frac{\pi}{6},\frac{2\pi}{3}$

$\theta=\frac{\pi}{6},\frac{2\pi}{3},\frac{\ 7\pi}{6},\frac{6\pi}{5}$

$\theta=\frac{\pi}{3},\frac{4\pi}{3},\frac{\ 7\pi}{3},\frac{12\pi}{5}$

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

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Solve the equation $2\sin\left(\theta\right)=\tan\left(\theta\right)$ for $0^{\circ}\le\theta\le360^{\circ}$ .
Note: Use the fact that $\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$ .

$\theta=0^{\circ},30^{\circ},90^{\circ},150^{\circ},180^{\circ}$

$\theta=0^{\circ},180^{\circ},360^{\circ}$

$\theta=0^{\circ},60^{\circ},180^{\circ},300^{\circ},360^{\circ}$

$\theta=0^{\circ},300^{\circ}$

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

For $0^{\circ}\le\theta\le360^{\circ}$ , solve $6\cos^2\left(\theta\right)+6\cos\left(\theta\right)-1=0$ .

$\theta=81.6^{\circ},120^{\circ},240^{\circ},278.4^{\circ}$

$\theta=81.6^{\circ},120^{\circ}$

$\theta=240^{\circ},120^{\circ}$

$\theta=81.6^{\circ},\ 278.4^{\circ}$

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

Solve the equation $\sin^2\left(\theta\right)+3\cos\left(2\theta\right)=2$ for $0\le\theta\le360\degree^{ }$ . Give your answer(s) to 1 decimal place.
Note: Use the fact that $\cos\left(2\theta\right)=1-2\sin^2\left(\theta\right)$

$\theta=153.4^{\circ}$

$\theta=26.6^{\circ}$

$\theta=26.6^{\circ},153.4^{\circ}$

$\theta=26.6^{\circ},153.4^{\circ},206.6^{\circ},333.4^{\circ}$

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A right-angled triangle XYZ has an angle, $\theta$ , where $\sin\left(\theta\right)=\frac{\sqrt{5}}{5}$ . Without evaluating $\theta$ , calculate the exact value(in surd form if applicable) of $\cos\left(\theta\right)$ .

$\cos\left(\theta\right)=2\sqrt{5}$

$\cos\left(\theta\right)=\frac{2}{5}$

$\cos\left(\theta\right)=\frac{\sqrt{5}}{5}$

$\cos\left(\theta\right)=\frac{2\sqrt{5}}{5}$

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A right-angled triangle XYZ has an angle, $\theta$ , where $\sin\left(\theta\right)=\frac{\sqrt{5}}{5}$ . Without evaluating $\theta$ , calculate the exact value(in surd form if applicable) of $\sin\left(2\theta\right)$ .
Note: Use the fact that $\sin\left(2\theta\right)=2\sin\left(\theta\right)\cos\left(\theta\right)$ .

$\sin\left(2\theta\right)=\frac{4}{5}$

$\sin\left(2\theta\right)=\frac{1}{5}$

$\sin\left(2\theta\right)=\frac{2\sqrt{5}}{5}$

$\sin\left(2\theta\right)=\frac{\sqrt{10}}{5}$

10.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A wire in the form of a circle with radius 4 cm is reshaped into the form of a sector of a circle with radius 10 cm. Determine, in radians, the angle of the sector, giving your answer in terms of $\pi$ .

$\theta=\frac{4\pi}{5}-1$

$\theta=\frac{2\left(2\pi-5\right)}{5}$

$\theta=\frac{4\pi}{5}$

$\theta=4\left(2\pi-5\right)$

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Trigonometry (Intermediate)

Which of the following graphs represent the function f(x)=cos⁡(2x)f\left(x\right)=\cos\left(2x\right)f(x)=cos(2x) ?

Which of the following is true?

Calculate the value(s) of θ\thetaθ such that cos⁡(θ)=−12\cos\left(\theta\right)=-\frac{1}{2}cos(θ)=−21​ for 0≤θ≤2π0\le\theta\le2\pi0≤θ≤2π .

Calculate the value(s) of θ\thetaθ such that tan⁡(2θ)=3\tan\left(2\theta\right)=\sqrt{3}tan(2θ)=3​ for 0≤θ≤2π0\le\theta\le2\pi0≤θ≤2π .

For 0∘≤θ≤360∘0^{\circ}\le\theta\le360^{\circ}0∘≤θ≤360∘ , solve 6cos⁡2(θ)+6cos⁡(θ)−1=06\cos^2\left(\theta\right)+6\cos\left(\theta\right)-1=06cos2(θ)+6cos(θ)−1=0 .

A right-angled triangle XYZ has an angle, \thetaθ , where sin⁡(θ)=55\sin\left(\theta\right)=\frac{\sqrt{5}}{5}sin(θ)=55​​ . Without evaluating θ\thetaθ , calculate the exact value(in surd form if applicable) of cos⁡(θ)\cos\left(\theta\right)cos(θ) .

A wire in the form of a circle with radius 4 cm is reshaped into the form of a sector of a circle with radius 10 cm. Determine, in radians, the angle of the sector, giving your answer in terms of π\piπ .

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Discover more resources for Mathematics

Which of the following graphs represent the function $f\left(x\right)=\cos\left(2x\right)$ ?

Calculate the value(s) of $\theta$ such that $\cos\left(\theta\right)=-\frac{1}{2}$ for $0\le\theta\le2\pi$ .

Calculate the value(s) of $\theta$ such that $\tan\left(2\theta\right)=\sqrt{3}$ for $0\le\theta\le2\pi$ .

For $0^{\circ}\le\theta\le360^{\circ}$ , solve $6\cos^2\left(\theta\right)+6\cos\left(\theta\right)-1=0$ .

A right-angled triangle XYZ has an angle, $\theta$ , where $\sin\left(\theta\right)=\frac{\sqrt{5}}{5}$ . Without evaluating $\theta$ , calculate the exact value(in surd form if applicable) of $\cos\left(\theta\right)$ .

A wire in the form of a circle with radius 4 cm is reshaped into the form of a sector of a circle with radius 10 cm. Determine, in radians, the angle of the sector, giving your answer in terms of $\pi$ .