# trigonometric identities problems

□​​. sin⁡2θ=12(1−cos⁡2θ)cos⁡2θ=12(1+cos⁡2θ).\begin{aligned} cot⁡π16⋅cot⁡2π16⋅cot⁡3π16×⋯×cot⁡7π16=(cot⁡π16⋅cot⁡7π16)⋅(cot⁡2π16⋅cot⁡6π16)⋅(cot⁡3π16⋅cot⁡5π16)⋅cot⁡4π16=(cot⁡π16⋅cot⁡(π2−π16))⋅(cot⁡2π16⋅cot⁡(π2−2π16))⋅(cot⁡3π16⋅cot⁡(π2−3π16))⋅cot⁡π4=(cot⁡π16⋅tan⁡π16)⏟1⋅(cot⁡2π16⋅tan⁡2π16)⏟1⋅(cot⁡3π16⋅tan⁡3π16)⏟1⋅1=1. \cot(-\theta) &= -\cot \theta\\ □​​. sin⁡2θ+cos⁡2θ=1cos⁡2θ=1−sin⁡2θ=1−1625=925⇒cos⁡θ=±35⇒cos⁡θ=−35. ... For a problem like sin(π/12), remember that θ/2 = π/12, or θ = π/6, when substituting into the identity. \csc \theta & = \sec \left( \frac{\pi}{2} - \theta \right).

Quotient Identities. Let A  =  (1 - cos2Î¸) csc2Î¸  and  B  =  1. \cos 2\theta &= \cos^2 \theta - \sin^2 \theta \\ & = 2 - 3 \\ □\begin{aligned} &=1 + 1 \\ □​​. \cot \dfrac{\pi}{16} \cdot \cot \dfrac{2 \pi}{16} \cdot \cot \dfrac{3 \pi}{16} \times \cdots \times \cot \dfrac{7 \pi}{16} Add rational expressions by finding common denominators. When working with trigonometric identities, it may be useful to keep the following tips in mind: The fundamental period of the graphs of sin⁡x,cos⁡x,csc⁡x,sec⁡x\sin x, \cos x, \csc x, \sec x sinx,cosx,cscx,secx is 2π,2\pi,2π, while the fundamental period of the graphs of tan⁡x,cot⁡x \tan x, \cot x tanx,cotx is π\piπ. Sitemap. Unit: Trigonometric equations and identities, Restricting domains of functions to make them invertible, Domain & range of inverse tangent function, Using inverse trig functions with a calculator, Solving sinusoidal equations of the form sin(x)=d, Trig word problem: solving for temperature, Level up on the above skills and collect up to 400 Mastery points, Proof of the sine angle addition identity, Proof of the cosine angle addition identity, Finding trig values using angle addition identities, Using trig angle addition identities: finding side lengths, Using trig angle addition identities: manipulating expressions, Find trig values using angle addition identities, Level up on the above skills and collect up to 200 Mastery points, Trig challenge problem: area of a triangle, Trig challenge problem: area of a hexagon, Trig challenge problem: cosine of angle-sum, Trig challenge problem: arithmetic progression, Trig challenge problem: multiple constraints, Trig challenge problem: system of equations. (2+1)(2−1)cos⁡θ=(2+1)sin⁡θ(2−1)cos⁡θ=2sin⁡θ+sin⁡θ⇒cos⁡θ−sin⁡θ=2sin⁡θ. Draw a picture illustrating the problem if it involves only the basic trigonometric functions. Our mission is to provide a free, world-class education to anyone, anywhere. □​​.

Trigonometric ratios of supplementary angles Trigonometric identities Problems on trigonometric identities Trigonometry heights and distances. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. A  =  (cos2Î¸/sin Î¸ cos Î¸) + (sin2Î¸/sin Î¸ cos Î¸), cos Î¸/(1 - tan Î¸) + sin Î¸/(1 - cot Î¸)  =  sin Î¸ + cos Î¸, Let A  =  cos Î¸/(1 - tan Î¸) + sin Î¸/(1 - cot Î¸)  and, A  =  cos Î¸/{1 - (sin Î¸/cos Î¸)} + sin Î¸/{1 - (cos Î¸/sin Î¸)}, A  =  cos2Î¸/(cos Î¸ - sin Î¸) + sin2Î¸/(sin Î¸ - cos Î¸), A  =  cos2Î¸/(cos Î¸ - sin Î¸) - sin2Î¸/(cos Î¸ - sin Î¸), A  =  (cos2Î¸ - sin2Î¸) / (cos Î¸ - sin Î¸), A  =  [(cos Î¸ + sin Î¸)(cos Î¸ - sin Î¸)] / (cos Î¸ - sin Î¸). \csc(-\theta) &= -\csc \theta\\ Trigonometric identities (trig identities) are equalities that involve trigonometric functions that are true for all values of the occurring variables. Pythagorean Identities. \end{aligned}cosθsinθcotθcscθ​=sin(2π​−θ)=cos(2π​−θ)=tan(2π​−θ)=sec(2π​−θ).​, sin⁡2θ=2sin⁡θcos⁡θcos⁡2θ=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θtan⁡2θ=2tan⁡θ1−tan⁡2θ.\begin{aligned} A  =  (sin Î¸/cos Î¸) â sin Î¸ + cos Î¸, (1 - cos Î¸)(1 + cos Î¸)(1 + cot2Î¸)  =  1.

â{(sec Î¸ â 1)/(sec Î¸ + 1)}  =  cosec Î¸ - cot Î¸. & = 2\big(1 - 3\sin^2 \theta \cos^2 \theta\big) - 3\big(1 - 2\sin^2 \theta \cos^2 \theta\big) \\ \cos \theta & = \sin \left( \frac{\pi}{2} - \theta \right) \\ If cos⁡θ+sin⁡θ=2cos⁡θ\cos \theta + \sin \theta = \sqrt{2} \cos \thetacosθ+sinθ=2​cosθ, find the value of cos⁡θ−sin⁡θ\cos \theta - \sin \thetacosθ−sinθ. Donate or volunteer today! The tree is 42.5 feet tall and the base of the tree is 28 feet away from the house. The derivations of the half-angle identities … \cot^2 \theta + 1 &= \csc^2 \theta. Let A  =  tan4Î¸ + tan2Î¸  and B  =  sec4Î¸ + sec2Î¸. $$\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$$, $$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$$, $$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$$, $$\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$$, $$\tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)}$$, $$\tan(a-b)=\frac{\tan(a)-\tan(b)}{1+\tan(a)\tan(b)}$$, $$\sin(x)+\sin(y)=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})$$, $$\sin(x)-\sin(y)=2\cos(\frac{x+y}{2})\sin(\frac{x-y}{2})$$, $$\cos(x)+\cos(y)=2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})$$, $$\cos(x)-\cos(y)=-2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})$$, $$\cos(2x)=\cos^2(x)-\sin^2(x)=1-2\sin^2(x) = 2\cos^2(x)-1$$, $$\sin(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{2}}$$, $$\cos(\frac{x}{2})=\pm\sqrt{\frac{1+\cos(x)}{2}}$$, $$\tan(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1+\cos(x)}$$, $$\sin(x)\cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}$$, $$\cos(x)\cos(y)=\frac{\cos(x+y)+\cos(x-y)}{2}$$, $$\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x+y)}{2}$$. \sin^2 \theta &=\frac{1}{2}(1-\cos 2\theta) \\

Let A  =  (1 - sin A)/(1 + sin A) and B  =  (sec A - tan A)2. New user? \cot \theta & = \tan \left( \frac{\pi}{2} - \theta \right) \\ □_\square□​. Check out all of our online calculators here! & = \dfrac{1}{\sin^2 \theta \cdot \cos^2 \theta} \\ \\ \end{aligned} sin2A+cos2Atan2A+1cot2A+1​===​1sec2Acsc2A.​, sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡Bcos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡Btan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B.\begin{aligned} \sin(A\pm B) &=& \sin A \cos B \pm \cos A \sin B \\ \cos(A\pm B) &=& \cos A \cos B \mp \sin A \sin B \\ \tan(A\pm B) &=& \frac{ \tan A \pm \tan B } { 1 \mp \tan A \tan B }. \big(\sqrt2 + 1\big)\big(\sqrt2 - 1\big)\cos\theta & = \big(\sqrt2 + 1\big)\sin \theta \\ \hline Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. Practice your math skills and learn step by step with our math solver. \tan(x+y) &= \dfrac{\tan x + \tan y}{1 - \tan x \tan y} \\ sin⁡2π10+sin⁡24π10+sin⁡26π10+sin⁡29π10=sin⁡2(π10)+sin⁡2(π2−π10)+sin⁡2(π2+π10)+sin⁡2(π−π10)=sin⁡2π10+cos⁡2π10⏟1+cos⁡2π10+sin⁡2π10⏟1=1+1=2. If xxx and yyy are acute angles such that. Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form. □\begin{aligned}

\end{aligned} sin2θcos2θtan2θ​=2sinθcosθ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ=1−tan2θ2tanθ​.​, sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡ysin⁡(x−y)=sin⁡xcos⁡y−cos⁡xsin⁡ycos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡ycos⁡(x−y)=cos⁡xcos⁡y+sin⁡xsin⁡ytan⁡(x+y)=tan⁡x+tan⁡y1−tan⁡xtan⁡ytan⁡(x−y)=tan⁡x−tan⁡y1+tan⁡xtan⁡y.\begin{aligned} Identities In Use‎ > ‎ Word Problem During a stormy night in Louisiana, a tree fell on a residents home. \end{aligned}(2​+1)(2​−1)cosθ(2−1)cosθ⇒cosθ−sinθ​=(2​+1)sinθ=2​sinθ+sinθ=2​sinθ. \cos(-\theta) &= \cos \theta \\ a^2 + 25 & = 25 \\ (tanθ+cotθ)2​=tan2θ+cot2θ+2tanθcotθ=tan2θ+cot2θ+2=(1+tan2θ)+(1+cot2θ)=sec2θ+csc2θ​, 2. sec⁡2θ+csc⁡2θ=1cos⁡2θ+1sin⁡2θ=sin⁡2θ+cos⁡2θsin⁡2θ⋅cos⁡2θ=1sin⁡2θ⋅cos⁡2θ=sec⁡2θ⋅csc⁡2θ. \end{aligned}cosxcosysinxcosycosxsinysinxsiny​=21​(cos(x−y)+cos(x+y))=21​(sin(x−y)+sin(x+y))=21​(sin(x+y)−sin(x−y))=21​(cos(x−y)−cos(x+y)).​, sin⁡x+sin⁡y=2sin⁡(x+y2)cos⁡(x−y2)cos⁡x+cos⁡y=2cos⁡(x+y2)cos⁡(x−y2).\begin{aligned} a^2 + 25 & = 9\sin^2 \theta + 16\cos^2 \theta + 24\sin\theta\cos\theta + 16\sin^2\theta + 9\cos^2\theta - 24\sin\theta\cos\theta \\ Trigonometric ratios of supplementary angles Trigonometric identities Problems on trigonometric identities Trigonometry heights and distances

By adding, (sec⁡θ+tan⁡θ)+(sec⁡θ−tan⁡θ)=23+32  ⟹  2sec⁡θ=136  ⟹  sec⁡θ=1312(\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = \frac23 + \frac32 \implies 2 \sec \theta = \frac{13}{6} \implies \sec \theta = \frac{13}{12}(secθ+tanθ)+(secθ−tanθ)=32​+23​⟹2secθ=613​⟹secθ=1213​. \\ Exercise 2 Knowing that tan α = 2, and that 180º < α < 270°, calculate the remaining trigonometric ratios of angle α. Exercise… Quotient Identities. \cos^2 \theta + \sin^2 \theta &= 1 \\ Let A  =  tan Î¸ sin Î¸ + cos Î¸  and B =  sec Î¸. Embedded content, if any, are copyrights of their respective owners. Let A  =  cot Î¸ + tan Î¸ and B  =  sec Î¸ csc Î¸. \end{aligned} sin2θcos2θ​=21​(1−cos2θ)=21​(1+cos2θ).​, cos⁡xcos⁡y=12(cos⁡(x−y)+cos⁡(x+y))sin⁡xcos⁡y=12(sin⁡(x−y)+sin⁡(x+y))cos⁡xsin⁡y=12(sin⁡(x+y)−sin⁡(x−y))sin⁡xsin⁡y=12(cos⁡(x−y)−cos⁡(x+y)).\begin{aligned}