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TrigonometryQuestion and Answers: Page 1

Question Number 208437    Answers: 1   Comments: 0

Question Number 208256    Answers: 0   Comments: 1

determinant ()

Question Number 208252    Answers: 1   Comments: 0

cos (((2π)/(21)))cos (((4π)/(21)))cos (((8π)/(21)))cos (((10π)/(22)))cos (((16π)/(21)))cos (((20π)/(21)))=?

$$\:\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{10}\pi}{\mathrm{22}}\right)\mathrm{cos}\:\left(\frac{\mathrm{16}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{20}\pi}{\mathrm{21}}\right)=? \\ $$

Question Number 208130    Answers: 1   Comments: 0

(1/(cos x−cos 3x)) + (1/(cos x−cos 5x)) + (1/(cos x−cos 7x)) + (1/(cos x−cos 11x))=?

$$\:\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{3x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{5x}}\:+ \\ $$$$\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{7x}}\:+\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{11x}}=?\: \\ $$

Question Number 208104    Answers: 0   Comments: 0

Question Number 207925    Answers: 1   Comments: 0

Question Number 207516    Answers: 2   Comments: 1

3(sinθ − cosθ)^4 + 6(sinθ + cosθ)^2 + 4(sin^6 θ + cos^6 θ) = ?

$$\mathrm{3}\left(\mathrm{sin}\theta\:−\:\mathrm{cos}\theta\right)^{\mathrm{4}} \:+\:\mathrm{6}\left(\mathrm{sin}\theta\:+\:\mathrm{cos}\theta\right)^{\mathrm{2}} \\ $$$$+\:\mathrm{4}\left(\mathrm{sin}^{\mathrm{6}} \theta\:+\:\mathrm{cos}^{\mathrm{6}} \theta\right)\:=\:? \\ $$

Question Number 207434    Answers: 0   Comments: 0

Relating to question 207407 x^3 −12x^2 +27x−17=0 Let x=t+4 t^3 −21t−37=0 The Trigonometric Solution gives these: x_1 =4−2(√7)cos ((π+2sin^(−1) ((37(√7))/(98)))/6) x_2 =4−2(√7)sin ((sin^(−1) ((37(√7))/(98)))/3) x_3 =4+2(√7)sin ((π+sin^(−1) ((37(√7))/(98)))/3) Prove these identities: x_1 =2−((1+2sin (π/(18)))/(2cos (π/9))) x_2 =2+((1+2cos (π/9))/(2cos ((2π)/9))) x_3 =((1+2(√3)sin ((2π)/9))/(2sin (π/(18))))

$$\mathrm{Relating}\:\mathrm{to}\:\mathrm{question}\:\mathrm{207407} \\ $$$${x}^{\mathrm{3}} −\mathrm{12}{x}^{\mathrm{2}} +\mathrm{27}{x}−\mathrm{17}=\mathrm{0} \\ $$$$\mathrm{Let}\:{x}={t}+\mathrm{4} \\ $$$${t}^{\mathrm{3}} −\mathrm{21}{t}−\mathrm{37}=\mathrm{0} \\ $$$$\mathrm{The}\:\mathrm{Trigonometric}\:\mathrm{Solution}\:\mathrm{gives}\:\mathrm{these}: \\ $$$${x}_{\mathrm{1}} =\mathrm{4}−\mathrm{2}\sqrt{\mathrm{7}}\mathrm{cos}\:\frac{\pi+\mathrm{2sin}^{−\mathrm{1}} \:\frac{\mathrm{37}\sqrt{\mathrm{7}}}{\mathrm{98}}}{\mathrm{6}} \\ $$$${x}_{\mathrm{2}} =\mathrm{4}−\mathrm{2}\sqrt{\mathrm{7}}\mathrm{sin}\:\frac{\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{37}\sqrt{\mathrm{7}}}{\mathrm{98}}}{\mathrm{3}} \\ $$$${x}_{\mathrm{3}} =\mathrm{4}+\mathrm{2}\sqrt{\mathrm{7}}\mathrm{sin}\:\frac{\pi+\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{37}\sqrt{\mathrm{7}}}{\mathrm{98}}}{\mathrm{3}} \\ $$$$\mathrm{Prove}\:\mathrm{these}\:\mathrm{identities}: \\ $$$${x}_{\mathrm{1}} =\mathrm{2}−\frac{\mathrm{1}+\mathrm{2sin}\:\frac{\pi}{\mathrm{18}}}{\mathrm{2cos}\:\:\frac{\pi}{\mathrm{9}}} \\ $$$${x}_{\mathrm{2}} =\mathrm{2}+\frac{\mathrm{1}+\mathrm{2cos}\:\frac{\pi}{\mathrm{9}}}{\mathrm{2cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}} \\ $$$${x}_{\mathrm{3}} =\frac{\mathrm{1}+\mathrm{2}\sqrt{\mathrm{3}}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}{\mathrm{2sin}\:\frac{\pi}{\mathrm{18}}} \\ $$

Question Number 207065    Answers: 1   Comments: 0

f(x)=[cos2x+cos3x][cos4x+cos6x][[cosx+cos5x] evaluar f(((2π)/(13)))

$$ \\ $$$$\:\:\:{f}\left({x}\right)=\left[{cos}\mathrm{2}{x}+{cos}\mathrm{3}{x}\right]\left[{cos}\mathrm{4}{x}+{cos}\mathrm{6}{x}\right]\left[\left[{cosx}+{cos}\mathrm{5}{x}\right]\right. \\ $$$${evaluar}\:\:\:{f}\left(\frac{\mathrm{2}\pi}{\mathrm{13}}\right)\:\: \\ $$

Question Number 206971    Answers: 1   Comments: 0

Construct an angle whose sine is (3/(2 + (√5))) .

$$\mathrm{Construct}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{whose}\:\mathrm{sine}\:\mathrm{is} \\ $$$$\frac{\mathrm{3}}{\mathrm{2}\:+\:\sqrt{\mathrm{5}}}\:. \\ $$

Question Number 206970    Answers: 2   Comments: 0

If sinθ = ((m^2 + 2mn)/(m^2 + 2mn + 2n^2 )) then prove that tanθ = ((m^2 + 2mn)/(2mn + 2n^2 )) .

$$\mathrm{If}\:\mathrm{sin}\theta\:=\:\frac{{m}^{\mathrm{2}} \:+\:\mathrm{2}{mn}}{{m}^{\mathrm{2}} \:+\:\mathrm{2}{mn}\:+\:\mathrm{2}{n}^{\mathrm{2}} }\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\mathrm{tan}\theta\:=\:\frac{{m}^{\mathrm{2}} \:+\:\mathrm{2}{mn}}{\mathrm{2}{mn}\:+\:\mathrm{2}{n}^{\mathrm{2}} }\:. \\ $$

Question Number 206899    Answers: 3   Comments: 0

If tanθ = ((2x(x + 1))/(2x + 1)) then find sinθ and cosθ.

$$\mathrm{If}\:\mathrm{tan}\theta\:=\:\frac{\mathrm{2}{x}\left({x}\:+\:\mathrm{1}\right)}{\mathrm{2}{x}\:+\:\mathrm{1}}\:\mathrm{then}\:\mathrm{find}\:\mathrm{sin}\theta\:\mathrm{and} \\ $$$$\mathrm{cos}\theta. \\ $$

Question Number 206833    Answers: 3   Comments: 0

Question Number 206787    Answers: 0   Comments: 1

Question Number 206727    Answers: 0   Comments: 0

Question Number 206618    Answers: 2   Comments: 0

If A = sin^4 θ + cos^4 θ then select the correct option: i) 0 < A < (1/2) ii) 1 < A < (3/2) iii) (1/2) ≤ A ≤ 1 iv) (3/2) ≤ A ≤ 2

$$\mathrm{If}\:\mathrm{A}\:=\:\mathrm{sin}^{\mathrm{4}} \theta\:+\:\mathrm{cos}^{\mathrm{4}} \theta\:\mathrm{then}\:\mathrm{select}\:\mathrm{the}\: \\ $$$$\mathrm{correct}\:\mathrm{option}: \\ $$$$\left.\mathrm{i}\right)\:\mathrm{0}\:<\:\mathrm{A}\:<\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{ii}\right)\:\mathrm{1}\:<\:\mathrm{A}\:<\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{iii}\right)\:\frac{\mathrm{1}}{\mathrm{2}}\:\leq\:\mathrm{A}\:\leq\:\mathrm{1} \\ $$$$\left.\mathrm{iv}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\leq\:\mathrm{A}\:\leq\:\mathrm{2} \\ $$

Question Number 206500    Answers: 2   Comments: 0

If asin^2 θ + bcos^2 θ = c, bsin^2 φ + acos^2 φ = d and atanθ = btanφ then prove that (1/a) + (1/b) = (1/c) + (1/d) .

$$\mathrm{If}\:{a}\mathrm{sin}^{\mathrm{2}} \theta\:+\:{b}\mathrm{cos}^{\mathrm{2}} \theta\:=\:{c},\:{b}\mathrm{sin}^{\mathrm{2}} \phi\:+\:{a}\mathrm{cos}^{\mathrm{2}} \phi\:=\:{d} \\ $$$$\mathrm{and}\:{a}\mathrm{tan}\theta\:=\:{b}\mathrm{tan}\phi\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:=\:\frac{\mathrm{1}}{{c}}\:+\:\frac{\mathrm{1}}{{d}}\:. \\ $$

Question Number 206471    Answers: 1   Comments: 0

If asinθ = bcosθ = ((2ctanθ)/(1 − tan^2 θ)) then prove that (a^2 − b^2 )^2 = 4c^2 (a^2 + b^2 ).

$$\mathrm{If}\:{a}\mathrm{sin}\theta\:=\:{b}\mathrm{cos}\theta\:=\:\frac{\mathrm{2}{c}\mathrm{tan}\theta}{\mathrm{1}\:−\:\mathrm{tan}^{\mathrm{2}} \theta}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\left({a}^{\mathrm{2}} \:−\:{b}^{\mathrm{2}} \right)^{\mathrm{2}} \:=\:\mathrm{4}{c}^{\mathrm{2}} \left({a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \right). \\ $$

Question Number 206434    Answers: 1   Comments: 0

If tan^2 θ = 1 − x^2 then prove that secθ + tan^3 θcosecθ = (√((2 − x^2 )^3 )) .

$$\mathrm{If}\:\mathrm{tan}^{\mathrm{2}} \theta\:=\:\mathrm{1}\:−\:{x}^{\mathrm{2}} \:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{sec}\theta\:+\:\mathrm{tan}^{\mathrm{3}} \theta\mathrm{cosec}\theta\:=\:\sqrt{\left(\mathrm{2}\:−\:{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:. \\ $$

Question Number 206421    Answers: 1   Comments: 0

If tanpθ = ptanθ then prove that ((sin^2 pθ)/(sin^2 θ)) = (p^2 /(1 + (p^2 − 1)sin^2 θ)) .

$$\mathrm{If}\:\mathrm{tan}{p}\theta\:=\:{p}\mathrm{tan}\theta\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{sin}^{\mathrm{2}} {p}\theta}{\mathrm{sin}^{\mathrm{2}} \theta}\:=\:\frac{{p}^{\mathrm{2}} }{\mathrm{1}\:+\:\left({p}^{\mathrm{2}} \:−\:\mathrm{1}\right)\mathrm{sin}^{\mathrm{2}} \theta}\:.\: \\ $$

Question Number 206362    Answers: 2   Comments: 0

sin(π/7) × sin((2π)/7) × sin((3π)/7) = ?

$$\mathrm{sin}\frac{\pi}{\mathrm{7}}\:×\:\mathrm{sin}\frac{\mathrm{2}\pi}{\mathrm{7}}\:×\:\mathrm{sin}\frac{\mathrm{3}\pi}{\mathrm{7}}\:=\:? \\ $$

Question Number 206048    Answers: 1   Comments: 0

Prove that 2^(sin^2 θ) + 2^(cos^2 θ) ≥ 2(√2).

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{2}^{\mathrm{sin}^{\mathrm{2}} \theta} \:+\:\mathrm{2}^{\mathrm{cos}^{\mathrm{2}} \theta} \:\geqslant\:\mathrm{2}\sqrt{\mathrm{2}}. \\ $$

Question Number 205808    Answers: 2   Comments: 3

Question Number 205596    Answers: 1   Comments: 1

Question Number 205535    Answers: 2   Comments: 0

Question Number 205429    Answers: 2   Comments: 0

If, ϕ = (1/2) (π −cos^( −1) ((1/4) )) ⇒ log_( 2) ( (( 1+ cos(6ϕ ))/(cos^6 (ϕ ))) ) =?

$$ \\ $$$$\:\mathrm{I}{f},\:\:\varphi\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\left(\pi\:−{cos}^{\:−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{4}}\:\right)\right) \\ $$$$ \\ $$$$\:\:\:\Rightarrow\:\mathrm{log}_{\:\mathrm{2}} \left(\:\frac{\:\mathrm{1}+\:{cos}\left(\mathrm{6}\varphi\:\right)}{{cos}^{\mathrm{6}} \left(\varphi\:\right)}\:\right)\:=? \\ $$$$ \\ $$

Question Number 205403    Answers: 1   Comments: 0

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