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: \(6+\frac{-6}{-x-3}=\frac{2 x}{x+3}+3 x\)
246
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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}}{...
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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 \(\theta\)
286
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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 ...
272
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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...
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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...
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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^{\...
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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...
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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...
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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
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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} \...
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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
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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
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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
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Show that \(\cos \left(90^{\circ}+\theta\right)=-\sin \theta\) by using the appropriate compound angle formula.
352
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Prove that the identity \(\cos (\mathrm{A}-\mathrm{B})=\cos \mathrm{A} \cdot \cos \mathrm{B}+\sin \mathrm{A} \cdot \sin \mathrm{B}\)
249
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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...
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Solve for \(x\) if \(\sin \left(x+30^{\circ}\right)=-0,2\) where \(x \in\left(-180^{\circ} ; 0^{\circ}\right)\)
394
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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}{\...
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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}...
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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
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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...
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