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Questions tagged [trigonometry]

Questions about trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles and other topics relating to measuring triangles.

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.

Trigonometry is most simply associated with planar right-angle triangles. The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles.

One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

See Wikipedia's list of trigonometric identities.

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Evaluating $\cos^{-1}\left(\,\sin\left(\pi^2\right)\,\right)$ without a calculator

This is one of the questions I got in my math exam: Evaluate without a calculator: $$\cos^{-1}\left(\,\sin\left(\pi^2\right)\,\right)$$ I just can't figure out how to evaluate $\sin(\pi^2)$ without using a calculator. Thanks!
hhalaweh
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For $x\in (0,\pi)$, how many solutions does the equation $\sin x+2\sin2x-\sin3x=3$ have?

For $x\in (0,\pi)$, how many solutions does the equation $\sin x+2\sin2x-\sin3x=3$ have? My attempt is as follows: One can see that if $\sin3x$ is positive, then it is not possible to have any solution, so $\sin3x$ should be…
user3290550
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Find the general solution of $\theta$ for which the following quadratic equation is the square of a linear function.

Find the general solution of $\theta$ for which the quadratic equation $$\left(\sin\theta\right)x^2+(2\cos\theta)x+\dfrac{\cos\theta+\sin\theta}{2}$$ is the square of a linear…
user3290550
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In the range $0\leq x \lt 2\pi$ the equation has how many solutions $\sin^8 {x}+\cos^6 {x}=1$

In the range $0\leq x \lt 2\pi$ the equation has how many solutions $$\sin^8 {x}+\cos^6 {x}=1$$ What i did $\cos^6 {x}=1-\sin^8 {x}=(1-\sin^4 {x})(1+\sin^4 {x})=(1-\sin^2 {x})(1+\sin^2 {x})(1+\sin^4 {x})$ $\cos^4 {x}=(1+\sin^2 {x})(1+\sin^4 {x}) ,…
AKA Death
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If $3\sec^4\theta+8=10\sec^2\theta$, find the values of $\tan\theta$

If $3\sec^4\theta+8=10\sec^2\theta$, find the values of $\tan\theta$. $$3\sec^4\theta-10\sec^2\theta+8=0$$ $$3\sec^4\theta-6\sec^2\theta-4\sec^2\theta+8=0$$ $$(3\sec^2\theta-4)(\sec^2\theta-2)=0$$ $$\sec^2\theta=2 \text { or }…
user3290550
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Write the expression in terms of $\sin$ only $\sin(4x)-\cos(4x)$

I am currently taking a Precalc II (Trig) course in college. There is a question in the book that I can't figure out how to complete it. The question follows: Write the expression in terms of sine only: $\sin(4x)-\cos(4x)$ So far I have…
BlazeRyder
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Proof: minimum volume of a notch cut at equal angle of cutting surfaces with horizontal plane

A notch is cut in a cylindrical vertical tree trunk. The edge of the cut reaches the axis of the cylinder and the cut is between two half-circle planes. Each half-circle is bounded by a horizontal line passing through the axis of the cylinder. The…
user706791
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Find minimum of $\sin^4 x + \cos^4 x + \sec^4 x$

I tried manipulating the terms but I couldn't get anywhere. The only other thing I can observe is that the minimum must be greater than $1$ since all the terms are non-negative and the range of $\sec^4 x$ is $[1, \infty)$. Also it can't be $1$ since…
Alraxite
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Different Trigonometric Equations have different general solutions

The NCERT Math Textbook for Grade 11 mentions these two general solutions for Cosine Trigonometric function: $\cos x = 0$ gives $x=(2n+1)\pi/2$, where $n\in Z$ $\cos x = \cos y$ gives $x=2n\pi \pm y$, where $n\in Z$ So if I have to solve $\cos…
Alekhya Vellanki
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if $0 < x <1$, How can I prove $ \ln{x} > 1- \frac{1}{x}$ without derivative or integral

If $0 < x <1$, how can I prove$$ \ln x > 1- \frac{1}{x}$$ without derivative or integral? In the process of proving $$\sin x ^ {\sin x}> \cos x ^ {\cos x}$$ without calculus, I had to solve the above equation, but the idea does not come to mind.
Dongmin Choi
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Why cant I prove this trigonometric equation straight down?

The question is as follows: Given that $\tan^2a - 2 \tan^2b = 1$. Show that $\cos2a + \sin^2b = 0$. After a few attempts, I successfully came up with a solution as follows: $$\tan^2a - 2 \tan^2b = 1$$ $$(\sec^2a - 1) - 2(\sec^2b - 1) = 1$$ $$\sec^2a…
Y.T.
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Trigonometric equations: cotangent

If I have $cot(x-a)=cot(x-b)$ Where x is in radians and equal on both the sides and not equal to $0$ or $π$ Also for a and b, they are not equal to $0$ or $π$ Does the above equality mean $a=b$? If not then how do we even find the value of a and…
Korra
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Find the expression of an angle within an equation that contain sin(angle) and cos(angle)

Let's consider the following equation: $$R = \frac{p D\cos\theta}{p - D \sin\theta}$$ Where $R,p,D>0$ and $-\pi/2\le\theta\le0$. The goal is to transform this formula in order to write something like $\theta = f(R,p,D)$ My first try was to write…
Temani Afif
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If $\sec A-\cos A=1$, then determine the value of $\tan^2\frac A2$

This is what I tried $\sec A=\frac{1}{\cos A}$, so the equation becomes $1-\cos^2A=\cos A$ If we solve the above quadratic equation, we the values of $\cos A$ as $\frac{-1\pm \sqrt5}{2}$ Therefore, $\tan\frac A2$ becomes $$\sqrt \frac{3-\sqrt…
Aditya
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