Recent questions tagged trigonometry

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Define the $l^2$ norm of $x \in \mathbb{R}^n$ to be$$\Vert x \Vert_2 = \sqrt{ \sum_{i=1}^n | x_i |^2}$$Prove that $\Vert \cdot \Vert_2$ is a norm.
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Simplify $$(\sec A+\tan A)(1-\sin A)=$$
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$$\operatorname{cosec}\left(90^{\circ}-\mathrm{A}\right)$$ is equal to
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$$\left(\cos \left(45^{\circ}\right)+i \sin \left(45^{\circ}\right)\right)^{4}=$$
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If $$\cos 34^{\circ}=t$$, express each of the following in terms of $$t$$ :1. $$\quad \sin 34^{\circ}$$2. $$\tan 146^{\circ}$$
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What is trigonometry for?
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Solve for $$\mathrm{x}$$ in the following equation.$\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x}=4$
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Prove that $$\tan x=\frac{\sin x}{\cos x}$$
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If $$\sin x-\cos x=0$$, then what is the value of $$\sin ^4 x+\cos ^4 x ?$$
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What is a zero-angle?
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$\tan (\theta)=\frac{\sqrt{145}-1}{12}$Find $$\tan (2 \theta)$$
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Solve for $$x$$ without using a calculator: $$\sqrt{4-x}+x=-2$$
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Solve for $$x$$ without using a calculator: $$6+\frac{-6}{-x-3}=\frac{2 x}{x+3}+3 x$$
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Solve for $$x$$ without using a calculator. $$\frac{1}{4} x^{\frac{3}{2}}=2$$
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Solve for $$x$$ without using a calculator. $$(9)^{x^2-9}=1$$
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Solve for $$x$$ without using a calculator. $$8^{-x}=\frac{1}{64}$$
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Simplify the following without a calculator:a) $$\sqrt{9 x^4+16 x^4}$$b) $$\left(\frac{b^{-\frac{1}{3}}}{\sqrt[3]{b^2}}\right)^{-2}$$c) $$\frac{9^{n-1} \cdot 27^{3-2 n}}{... 234 views 1 answers Solve for \(x$$ in $$2 x^2 \geq 18$$
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If $$\sin \theta = \frac{12}{13}$$,find the other five trigonometric function values for $$\theta$$
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If $$\tan \theta=\frac{3}{4}$$ and $$\theta \in\left(180^{\circ} ; 360^{\circ}\right)$$ calculate without the use of a calculator and with the aid of a diagram the value ...
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(a) Show that $$\sin \left(45^{\circ}-\alpha\right)=\dfrac{\sqrt{2}(\cos \alpha-\sin \alpha)}{2}$$(b) Hence prove that $$\sin 2 \alpha+2 \sin ^2\left(45^{\circ}-\alpha\ri... 368 views 0 answers Calculate: \(\sin 110^{\circ}+\cos 70^{\circ} \cdot \tan 190^{\circ}$$
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Evaluate without using a calculator: $$\dfrac{\sin 72^{\circ}}{\sin 24^{\circ}}-\dfrac{\cos 72^{\circ}}{\cos 24^{\circ}}$$
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What are the compound angle formulae?
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Show that $$\sin 8 \theta=8 \sin \theta \cdot \cos \theta \cdot \cos 2 \theta \cdot \cos 4 \theta$$
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Simplify the following without using a calculator:(a) $$2 \sin 22,5^{\circ} \cdot \cos 22,5^{\circ}$$(b) $$2 \sin 75^{\circ} \cdot \cos 75^{\circ}$$(c) $$2 \sin \left(-15... 241 views 0 answers Expand the following using double angle formulae:(a) \(\cos 4 x$$(b) $$\sin 6 \theta$$(c) $$\cos 10 \theta$$(d) $$\sin 50^{\circ}$$(e) $$\cos 20^{\circ}$$(f) $$\sin 70^{\... 674 views 1 answers Calculate the following without using a calculator:(a) \(2 \sin 15^{\circ} \cdot \cos 15^{\circ}$$(b) $$1-2 \cos ^2 22,5^{\circ}$$(c) $$\left(\cos 15^{\circ}+\sin 15^{\ci... 430 views 1 answers Write as the cosine of a single angle:(1) \(\cos ^2 2 \theta-\sin ^2 2 \theta$$(2) $$1-2 \sin ^2 2 \theta$$(3) $$\sin ^2 \theta-\cos ^2 \theta$$
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Write as the sine of a single angle:(1) $$2 \sin 3 \theta \cdot \cos 3 \theta$$(2) $$2 \sin \frac{\mathrm{A}}{2} \cdot \cos \frac{\mathrm{A}}{2}$$(3) $$4 \sin x \cos x$$
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Use the double angle formulae of cosine to expand the following:(1) $$\cos 6 \mathrm{~A}$$(2) $$\cos 80^{\circ}$$(3) $$\cos \mathrm{A}$$
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Use the double angle formulae of sine to expand the following:(1) $$\sin 4 \mathrm{~A}$$(2) $$\sin 50^{\circ}$$(3) $$\sin \mathrm{A}$$
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Show that if $$\sin \left(\theta+60^{\circ}\right)=2 \sin \theta$$ then $$\tan \theta=\dfrac{\sqrt{3}}{3}$$
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Calculate the following without the use of a calculator:(a) $$\cos 75^{\circ}$$(b) $$\sin 15^{\circ}$$(c) $$\cos 165^{\circ}$$
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Prove the following:(a) $$\cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cdot \cos \beta$$(b) $$\sin \left(\theta+30^{\circ}\right)-\sin \left(\theta-30^{\circ}\r... 597 views 1 answers Calculate \(\sin 75^{\circ}$$ without the use of a calculator.
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(a) Show that $$\cos \left(60^{\circ}+\theta\right)-\cos \left(60^{\circ}-\theta\right)=-\sqrt{3} \sin \theta$$(b) Hence evaluate $$\cos 105^{\circ}-\cos 15^{\circ}$$ wit...
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Show that the following statements are identities using your compound-angle formulae: (a) $$\sin \left(180^{\circ}+\theta\right)=-\sin \theta$$ (b) $$\cos (\mathrm{A}-\ma... 410 views 0 answers Calculate each of the following without the use of a calculator:(a) \(\cos 80^{\circ} \cdot \cos 10^{\circ}-\sin 80^{\circ} \cdot \sin 10^{\circ}$$(b) $$\cos 70^{\circ} \... 192 views 0 answers Use the compound angle formulae to simplify each expression to one term only:(a) \(\cos 3 \theta \cdot \cos \theta+\sin 3 \theta \cdot \sin \theta$$(b) $$\cos 2 x \cdot \... 415 views 0 answers Expand each of the following using the compound angle formulae:(a) \(\cos \left(x-40^{\circ}\right)$$(b) $$\sin \left(x-50^{\circ}\right)$$(c) $$\cos (2 x+y)$$(d) $$\sin ... 449 views 1 answers Without using a calculator, calculate the value of each of the following:(1) \(\cos 320^{\circ} \cdot \cos 20^{\circ}-\sin 140^{\circ} \cdot \sin 200^{\circ}$$(2) $$\cos ... 290 views 1 answers Show that \(\cos \left(90^{\circ}+\theta\right)=-\sin \theta$$ by using the appropriate compound angle formula.
Prove that the identity $$\cos (\mathrm{A}-\mathrm{B})=\cos \mathrm{A} \cdot \cos \mathrm{B}+\sin \mathrm{A} \cdot \sin \mathrm{B}$$
Given:(i) $$\mathrm{A}=60^{\circ}$$ and $$\mathrm{B}=30^{\circ}$$(ii) $$\mathrm{A}=110^{\circ}$$ and $$\mathrm{B}=50^{\circ}$$(iii) $$\mathrm{A}=225^{\circ}$$ and $$\math... 372 views 0 answers Solve for \(x$$ if $$\sin \left(x+30^{\circ}\right)=-0,2$$ where $$x \in\left(-180^{\circ} ; 0^{\circ}\right)$$
Prove that the following statements are identities:(a) $$(\sin \theta+\cos \theta)^2=1+2 \sin \theta \cdot \cos \theta$$(b) $$\dfrac{1}{\tan \theta}=\dfrac{\cos \theta}{\... 365 views 0 answers If \(\tan 22^{\circ}=t$$ write the following in terms of $$t$$.(a) $$\tan 202^{\circ}$$(b) $$\tan 338^{\circ}$$(c) $$\tan \left(-22^{\circ}\right)$$(d) $$\tan 518^{\circ}... 325 views 0 answers Point \(\mathrm{M}(-5 ;-12)$$ is a point on the Cartesian plane and reflex of $$\mathrm{XO} M=\theta$$.Calculate the following without the use of a calculator. (a) $$\cos... 328 views 1 answers If \(\cos \mathrm{A}=\frac{2 \sqrt{6}}{5}$$ and $$\mathrm{A} \in\left[90^{\circ} ; 360^{\circ}\right]$$ calculate without the use of a calculator and with the aid of a di...