Trigonometry: Pythagorean Identity
Read this topicExplanations 4
- The Pythagorean Theorem leads to a simple proof of the Pythagorean Identity using only the basic definitions of trig functions. The Pythagorean Theorem on a right triangle with …
By Rajiv Movva
- Suppose you have the expression cos2(x)+sin2(x)\cos^2(x)+\sin^2(x)cos2(x)+sin2(x). Let's think about how we would simplify it. One way to represent cos(x)\cos(x)cos(x) and si…
By Richard Qi, Anusha Rahman
- (Video) Pythagorean Identities - Examples & Practice Problems, Trigonometry by The Organic Chemistry Tutor Check out this video by the Organic Chemistry Tutor to learn about the…
By Trisha, Anusha Rahman
- Trigonometry: Pythagorean Identity Now that we've covered the tangent identity and reciprocal identities, we can move on. In this lesson, we are going to learn about the Pythago…
By Trisha, Anusha Rahman
Problems 26
- Solve the following equation for xxx in the interval [−π2,π2]:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]:[−2π,2π]: sinx+sin2x=cos2x.\sin x + \sin^2 x = \cos^2 x.sinx+sin2x=…
- Let xxx be a real number such that secx−tanx=2\sec x - \tan x = 2secx−tanx=2. Then secx+tanx=\sec x + \tan x =secx+tanx= (1999 AHSME #15)
- Given an acute angle AAA with sinA=1213\sin A = \frac{12}{13}sinA=1312, compute cosA\cos AcosA.
- If sin(θ)=45\sin(\theta) = \frac{4}{5}sin(θ)=54 and xcos(θ)=9x \cos(\theta) = 9xcos(θ)=9, find xxx.
- 32sin(x)=cos2(x)\frac{3}{2}\sin(x) = \cos^2(x)23sin(x)=cos2(x) If −π2≤x≤π2-\frac{\pi}{2} \le x \le \frac{\pi}{2}−2π≤x≤2π , what is πx\frac{\pi}{x}xπ? Hint: Remember, sin2…
- Let x=sin(A)x = \sin(A)x=sin(A). Using the general angle sum identities and the double angle identities, find sin(3A)\sin(3A)sin(3A) in terms of xxx.
- If tanx=52\tan x = \frac{\sqrt5}{2}tanx=25, what is secx\sec xsecx if 0<x<π0 < x < \pi0<x<π?
- Let xxx satisfy sin(x)=2−2\sin(x) = \sqrt{2 - \sqrt{2}}sin(x)=2−2. What is tan(x)\tan(x)tan(x)? (Hint: Use an identity to find cos(x)\cos(x)cos(x) first.)
- Find all solutions on the interval [0,2π)[0,2\pi)[0,2π) for 8cos2(θ)=3−2cos(θ).\\8\cos^2(\theta)=3-2\cos(\theta).8cos2(θ)=3−2cos(θ).
- cos(x)2+sin(x)2=?\cos\left(x\right)^2+\sin\left(x\right)^2=?cos(x)2+sin(x)2=?
- Simplify: sin4x−cos4x\sin^{4}x - \cos^{4}xsin4x−cos4x
- Simplify: cos2x+cos2xtan2x\cos^{2}x + \cos^{2}x \tan^{2}xcos2x+cos2xtan2x
- Simplify: 1−cos2xsec2x−1\dfrac {1-\cos^{2}x}{ \sec^{2}x -1}sec2x−11−cos2x
- In a triangle, if we have sinx=45\sin x = \dfrac {4}{5}sinx=54, what is the value of cosx\cos xcosx?
- Simplify the following expression to a single function: (1−cos2x)(cscx)(1-\cos^{2}x)(\csc x)(1−cos2x)(cscx)
- In a triangle, suppose sinx=0.25\sin x = 0.25sinx=0.25. What is cosx\cos xcosx?
- Simplify: (1–sin(x))(1+sin(x))(1 – \sin(x))(1 + \sin(x))(1–sin(x))(1+sin(x)).
- Simplify: cos2x+cos2xtan2x\cos^{2} x + \cos^{2}x \tan ^{2} xcos2x+cos2xtan2x
- Simplify: sin2x1−sin2x\dfrac {\sin^{2} x} {1-\sin^{2} x}1−sin2xsin2x
- Write sinx\sin xsinx in terms of cosx\cos xcosx.
- If sinx=35\sin x = \dfrac{3}{5}sinx=53, solve for cosx\cos xcosx.
- Solve cos3(α)−sin3(α)=0\cos^3(\alpha) - \sin^3(\alpha) = 0cos3(α)−sin3(α)=0 on the interval [0,π].[0, \pi].[0,π].
- Solve the following equation in x∈[0,2π]:x \in [0, 2\pi]:x∈[0,2π]: sin2(x)+cos2(x)+tan2(x)=csc2(x).\sin^2(x) + \cos^2(x) + \tan^2(x) = \csc^2(x).sin2(x)+cos2(x)+tan2(x)=csc2…
- If xxx is a real number for which cos(2x)=710,\cos(2x) = \frac{7}{10},cos(2x)=107, find the value of sin2(x).\sin^2(x).sin2(x).
- If xxx is a real number such that cos(2x)=710,\cos(2x) = \frac{7}{10},cos(2x)=107, find the value of cos2(x).\cos^2(x).cos2(x).
- Find the value of aaa for which tan2(x)−a=sec2(x)\tan^2(x) - a = \sec^2(x)tan2(x)−a=sec2(x) for all real numbers x.x.x.