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

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2
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3answers
44 views

Exact value for $\cos 36°$

Good morning! I'm having trouble with this problem... It's just taking me forever and I'm worn out and I'm lost on how to use a double angle identity for $72=2⋅36$ The problem reads as follows An ...
0
votes
0answers
15 views

What is the Winding Function? [on hold]

I've often heard of a mnemonic device called "SOH-CAH-TOA" used to learn about sine, cosine and tangent. But many of my math peers tell me that this device is not very good because it doesn't give an ...
0
votes
0answers
15 views

combine $\cos2t+\sqrt{\sin2t}$ a single wave of form $A \cos (wt-\theta)$

Combine $\cos2t+\sqrt{\sin(2t)}$ in a single wave of form $$A \cos {(wt-\theta)}$$ Hence plot arough sketch of the graph of the wave
0
votes
1answer
9 views

Finding angles in Barycentric system

How to find the angles of a triangle given the barycentric coordinates of its corners? Does it work if i take the first two components of every coordinate, and find the angles in the triangle (on the ...
3
votes
0answers
41 views

solving complex trig functions

How to solve: $$\sin(2x) + \sin(3x) = \frac{\sqrt{3}}{2}$$ where $x$ is in $[-\pi,\pi]$? I have no idea what to do with the $\sin(2x) + \sin(3x)$. Am I supposed to factorise, differentiate, is ...
1
vote
1answer
28 views

sum of an arctan series using mathematical induction

How to solve this problem using mathematical induction: $$\arctan (1) + \arctan \Big(\frac13\Big) + ... + \arctan \bigg(\frac{1}{n^2+n+1}\bigg)=\arctan (n+1)$$
0
votes
1answer
14 views

Find term for one angle of two in a trig function

In a right angled triangle, I know that $\tan (x) = \cfrac{4}{z}$ and that $\tan(x+y) = \cfrac{12}{z}$. I need to find an equation which has only $\tan(y)$. The answer is $\cfrac{12}{z} = ...
1
vote
4answers
42 views

Prove this identity: $\sin^4x = \dfrac{1}{8}(3 - 4\cos2x + \cos4x)$.

The problem reads as follows. Prove this identity: $\sin^4x = \dfrac{1}{8}(3 - 4\cos2x + \cos4x)$. I started with the right side and used double angles identities for $\cos2x$ and a sum and then ...
0
votes
2answers
26 views

If $\sin B = −1/2$ with $B$ in QIII, find $\cos B/2$

For the following, assume that all the given angles are in simplest form, so that if A is in QIV you may assume that 270° < A < 360°. If $\sin B = −1/2$ with B in QIII, find $\cos B/2$ Here's ...
0
votes
1answer
14 views

If $\sin A = 4/5$ with $A$ in QII, find $\cos A/2$

For the following, assume that all the given angles are in simplest form, so that if A is in QIV you may assume that 270° < A < 360°. If $\sin A = 4/5$ with A in QII, find $\cos A/2$ I keep ...
0
votes
1answer
33 views

If $\sin (B) = − \frac 1 2 $ with $B$ in third quadrant, then find $\cot (B/2)$

If $\sin (B) = − \frac 1 2 $ with $B$ in third quadrant, then find $\cot (B/2)$ I'm getting $-\sqrt{3}-2$
1
vote
1answer
46 views

Prove $\cos 3\theta = 4 \cos^3\theta − 3 \cos \theta$

$\cos 3θ = 4 \cos^3 θ − 3 \cos θ$ Here's my attempt. Is it correct? Thanks! $\cos(3θ)$ $= \cos(2θ + θ)$ $= \cos(2θ)\cos(θ) - \sin(2θ)\sinθ$ $= (2\cos^2θ - 1)\cosθ - (2\sinθ\cosθ)\sinθ$ $= ...
12
votes
9answers
2k views

Are there 3 trig functions or are there 6 trig functions?

In my algebra class we are being taught that there are only the 3 basic trig functions (cosine, sine, and tangent). But my friend who is 2 math grade levels ahead of me is saying that there is 6 trig ...
1
vote
0answers
16 views

Would every half angle of an angle in each quadrant be in the previous quadrant?

For example, take (5pi)/4 which is in Q3, it's half angle is (5pi)/8 which is in Q2. Is this true for every angle?
2
votes
1answer
21 views

Prove this identity: $ \tan(2x)-\sec(2x) =\tan(x-\pi/4)$

I've been having a time with this problem. I tried to start with the left side but I hit a dead end quick... I then tried the right side and had a little more luck but I've hit a block. I first used ...
1
vote
3answers
63 views

Range of f(x) = $\frac{\sqrt3\,\sin x}{2 + \cos x}$ [duplicate]

Can you give any idea about the range of the following function? $$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$
0
votes
3answers
41 views

Prove this identity? $\cos t ⋅ \cos u ⋅ \cos v = \frac14(\cos(t + u + v)+ \cos(t + u - v)+cos(t-u-v))$

The problem reads as follows. Prove the identity $$\cos t⋅\cos u⋅\cos v =\frac14\big(\!\cos(t + u + v)+\cos(t + u - v)+\cos(t-u-v)\big)$$ Hint: begin with the right side and use cosine sum identity ...
0
votes
2answers
42 views

Show that $(1 – \cos θ – \sin θ )^2 – 2(1 – \sin θ )(1 – \cos θ ) = 0$.

Show that $(1 – \cos θ – \sin θ )^2 – 2(1 – \sin θ )(1 – \cos θ ) = 0$. What kind of formulas should I use?
2
votes
0answers
32 views

($\cos^4x$)($\sin^2x$) in terms of first power of cosine

I believe that I have his correct but if someone could check it and see that'd be great. Here's a pic! [IMG]http://i58.tinypic.com/2dgm5ic.jpg[/IMG]
-1
votes
1answer
29 views

Find the value of $\theta$?

An operation maps the point $(x, y)$ on to the point $(x cos \theta, y sin \theta)$. i) Find the value of $\theta$ for which the y-axis is the image of the line $y = x$. ii) Draw a diagram to show ...
1
vote
2answers
50 views

Trig and Radians Confusion

I am learning about radians in my current class and am totally confused. How does $\sin(x+\frac\pi 2)=\cos(x)$ when $\frac\pi 2<x$ < $\pi$. I drew the triangles and I got $\sin(x+\frac\pi ...
0
votes
2answers
312 views

Ferris wheel question from Checkpoint book 11-14

The question is from checkpoint book 11-14 from section 4, chapter 19, Shape, Space and measures. Question number 8 A ferris wheel, centre O, has a diameter of 10m and carries eight equally spaced ...
0
votes
4answers
55 views

Is $\sin(\arcsin(x))$ equal to $x$?

I have a question. Is $\arcsin(\sin (x))$ or $\sin(\arcsin(x))$ always equal to $x$? And also for all other trigonometric ratios?
0
votes
1answer
31 views

Hint finding exact value of half-angle when $\tan (\theta) = {3}$

Unlike others I've tried, I'm having a hard time with this half-angle exercise: If $tan(\theta)={3}$ and $\theta$ is in QIII, find $\tan\left(\frac{\theta}{2}\right)$ Here's what I know (or think I ...
0
votes
4answers
33 views

Multiple trigonometric functions

How can you solve such a problem where multiple trigonometric functions are applied? Find the value of $\sin(\text{arc}\cot(\tan(\arccos\frac{3}{\sqrt{13}})))$.
0
votes
0answers
9 views

Get Attitude from 2-axis vector

I've built a quadrotor but my 3-axis accelerometer has a fault, the Z-Axis doesn't work. I would normally get my attitude with the following code pitch = atan2(accel_X, accel_Z)*RadToDeg; roll= ...
10
votes
2answers
355 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
0
votes
1answer
30 views

Writing expressions in terms of only sine

If I were to do this without these formulas, I would pull out a number that made both of the numbers(like (sqrt(3))/2 and 1/2) in the picture would be something that I could get a sine and cosine that ...
17
votes
1answer
443 views

Evaluating $\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$

How to integrate? $$\int{ \frac{\arctan\sqrt{n^{2}-1}}{\sqrt{n^{2}+n}}} dn$$ I have no idea how to do it. Tried to get some information from wiki, but its too hard :|
0
votes
0answers
7 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
0
votes
2answers
35 views

Was I wrong to omit angles in the solution set for this multiple angle problem?

I may have missed this in my precalculus course, but why was I wrong to omit angles that did not have a positive value for cosine? I didn't include $\frac{3\pi}{4},\frac{7\pi}{12},\frac{5\pi}{4}$ ...
0
votes
1answer
23 views

Different ways to formally define trigonometric functions

When I first learnt trigonometric functions I was in highschool and obviously the explanation they gave me was mostly intuitive. Now that I have taken my first curse of calculus I learnt a formal ...
1
vote
1answer
20 views

Check my solution to this trig inequality

Problem $1.88$ : Solve $$\cos x\lt \frac{\sqrt{3}}{2},\qquad x \in [0,2\pi]$$ I found the set of solutions to be $S=[0,2\pi]-\left[\dfrac{\pi}{6},\dfrac{11\pi}{6}\right]$ Is this correct? Thank you.
0
votes
4answers
36 views

Prove that $\sin(\frac{\pi}{3}+x)=\cos(\frac{\pi}{6}-x)$

How to prove that $\sin(\frac{\pi}{3}+x)=\cos(\frac{\pi}{6}-x)$ without using calculus just trigonometry?
0
votes
3answers
17 views

Find in terms of $p$, $\tan(-\alpha)$, $\tan(\pi - \alpha)$ and $\tan(\frac{\pi}{2}-\alpha)$.

Given that $\tan$ $\alpha = p$, where $\alpha$ is acute, find in terms of $p$, $\tan(-\alpha)$, $\tan(\pi - \alpha)$ and $\tan(\frac{\pi}{2}-\alpha)$.
2
votes
3answers
66 views

$\tan(x)=\cot(90^\circ-x)$??

I was looking at a mark scheme for a question I was stuck on, and I came across this. You are asked to work out the value of $\tan 75^\circ$ after you've worked out $\cos 15^\circ$ and $\sin ...
0
votes
4answers
39 views

A trigonometric equality

Can you help me prove: $\arccos \frac{y}{\sqrt{y^2 + x}} = \mathrm{arccot} \frac{y}{\sqrt{x}}$? I could solve this problem myself, but maybe you can show me a simple way to prove this and similar ...
1
vote
1answer
17 views

Intersection of graphs, and no solution for trig functions.

All I know is the c=asin(x-b) I don't know how to check the values of b for 'no solutions,' in the case of trig functions. Can someone people provide an algebraic method to solve this.
4
votes
2answers
122 views

In triangle, $\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$

To prove $$\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$$ My approach : $$ \begin{align} \text{L.H.S.} & = ...
7
votes
10answers
2k views

Prove $\sin^2\theta + \cos^2\theta = 1$

How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$ I'm curious to know of the different ways of proving this depending on different characterizations of sine and ...
-1
votes
1answer
63 views

Prove that $\cos^2\theta+\sin^2\theta=1$ [duplicate]

I try to find the question but I didn't How do you do it? I'm really stuck on this proof. Can someone please explain?
0
votes
2answers
60 views

How to solve this equation: $x+2 \tan(x)=\pi/2$

By drawing graph,or otherwise,find the number of roots of the equation $x+2 \tan(x)= \pi/2$ lying between $0$ and $2\pi$, and find the approximate value of the largest root. I found 3 roots ...
-2
votes
1answer
24 views

Help on Quadratic Equations [on hold]

If $\sin15$ and $\cos 15$ are the roots of a quadratic equation $x^2+ax+b=0$, then find the value of $a^4- b^2$. Please, need help, show working, thanks.
7
votes
3answers
3k views

Is $\tan(\pi/2)$ undefined or infinity?

The way I have understood, $0/0$ is undefined or indeterminate because, if $c=0/0$ then $c\cdot 0=0$, where $c$ can be any finite number including $0$ itself. If we also observe a fraction $F=a/b$ ...
0
votes
1answer
32 views

Real world tangent functions

I am a high school math teacher and one of my students asked me for examples of real world tangent functions. Not using tangent to find a side length but a relationship that can be represented by a ...
1
vote
2answers
51 views

Fourier Transform the following exponential and cosine function: $f(x) = e^{-a^{2}x^{2}}cos(bx)$

I have a previous exam here for my course (Provided by the professor) that requires me to do a Fourier Transform of the following equation. Here is the function: $f(x) = e^{-a^{2}x^{2}}cos{(bx)}$ ...
2
votes
4answers
103 views

Why do you have to begin with the largest angle or side when using law of cosines

Explain why you should always start with the largest angle or the largest side when using law of cosines. I don't understand why but my professor says so.
19
votes
7answers
3k views

How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?

I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?
1
vote
1answer
34 views

Solve the equation given below…

I have such an exercise: $$\color{teal}{{|x|\over{x}}\sin^2x-\cos|x|\cos x=1} $$ What I did is so: If $x\ge 0$ then we have: $$\sin^2x-\cos^2x=1$$ $$\sin^2x=1$$ So: $$\sin x=1$$ or $$\sin ...
1
vote
2answers
37 views

Trig identity question

Show that $\sin(2nx)=\sin((2n+1)x)\cos(x)-\cos((2n+1)x)\sin(x)$. I have the mark scheme in front of me, but it doesn't make sense to me... ...