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

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4k views

How to find the period of the sum of two trigonometric functions

I want to know if there exists a general method to find the period of the sum of two periodic trigonometric function. Example: $$f(x)=\cos(x/3)+\cos(x/4).$$
2
votes
1answer
77 views

What trig identities must one know to derive the others?

My TA told me in problem section one day that every trig identity could be derived from just 2: the Pythagorean identity and the double-angle identity (or he might have said the half-angle identity). ...
0
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0answers
29 views

Steps involved in simplifying trigonometric identities.

I am trying to master the simplification of trigonometric identities. When I look at a problem, asking me to simplify a trigonometric expression, I am not really sure what to do - but I do sort of ...
2
votes
2answers
142 views

How am I supposed to work this out, or do I have to memorize?

When simplifying a trigonometric expression, say, $\sin^2 \theta$ / $\cos^2 \theta$ - I remember that sin over cos is equal to tan. However, what other identities, such as the one mentioned above, ...
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1answer
67 views

Trigonometry - SSS formula to calculate angle of Triangle

I am trying to calculate angle using Law of cosines, SSS formula of Triangle. That is given three sides of triangle, calculate angle between sides. When there is large difference between all sides ...
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1answer
234 views

Conversion of angle from 360 degree to-90 degree

Here i am trying to convert angle into +90 degree AND -90 DEGREE FORMAT.For desired elevation angle i got answer properly.How to converert angle -90deg to zero, zero to +90 degree format. you can ...
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7answers
210 views

Value of $\cos^2\alpha-\sin^2\alpha$

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 8: b) If $\tan\alpha=r$, write an expression in terms of $r$ that represents the value of $\cos^2\alpha-\sin^2\alpha$. ...
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1answer
61 views

Integer Solutions to Cosine's Dot Product Formula

Say one wanted to test their students on the dot product formula without a calculator. One would (being a nice teacher and all) natural like to pick numbers in the plane that are "nice" and satisfy a ...
4
votes
1answer
139 views

Websites for math tests/quizzes [closed]

Next semester I'm taking calculus at college and I was looking for websites that have quizzes/test for things like trigonometry, trig formulas, pre-calculus, calculus readiness, etc. so I can get ...
1
vote
2answers
81 views

Trignometric functions, Maximum value

$f(x)=\sin^{16}(x) +\cos^{18}(x)$, find the maximum value of the given function $f(x)$. I tried differentiating the given function however couldn't get the value of $x$ for which I could substitute ...
3
votes
2answers
424 views

Evaluate integral by completing the square and doing trigonometric substitution

$\int \frac{1}{(x-2)\sqrt{x^{2}-4x+3}} dx$ is my problem Complete the square $\int \frac{1}{(x-2)\sqrt{(x-2)^{2}-1}} dx$ I know I'm probably supposed to use $ \frac{d}{dx}\operatorname{arcsec}(u) = ...
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2answers
96 views

How to solve $3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$

I have got a bunch of trig equations to solve for tomorrow, and got stuck on this one. Solve for $\theta$: $$3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$$ I tried using ...
2
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5answers
707 views

Period of $\sin(x) + \cos(x)$

The period of $\sin(x)$ is $2\pi$ and $\cos(x)$ is $2\pi$. And the period of $\sin(x)+\cos(x)$ is also $2\pi$. Why it is so?
3
votes
2answers
199 views

How to find the maximum value of $12\sin x -9\sin^2x$

How to find the maximum value of $12\sin x -9\sin^2x$ ; My approach : This can be written as $-[(3\sin x -2)^2-4]$. It means that the function will be maximum when $(3\sin x-2)^2 <4$ due to ...
2
votes
3answers
584 views

Deriving the sum-to-product identities

I've been asked by my textbook to derive the "sum-to-product" identities from the "product-to-sum" identities. I've attempted to to do this but i've met a dead end, and i'm quite confused. Using ...
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vote
2answers
346 views

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula.

Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. (a) $\cos^2 \left( \cfrac{θ}{2} \right)− \sin^2 \left( \cfrac{θ}{2} \right)$ (b) $2 \sin \left( ...
3
votes
1answer
115 views

Range of $f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}$ for a specified domain

We are asked to find the range of the function $$f(x)=\frac{\sin x -1}{\sqrt{3-2\cos x-2\sin x}}, \;\;\text{for}\;0\le x\le2\pi$$ I tried to find the range of each basic function of cos and sin then ...
3
votes
1answer
205 views

Solving this trigonometric equation

$$\sqrt{3} \cos x - 3 \sin x = 4 \sin 2x \;\cos 3x$$ I tried many things: opening $\sin 2x$, $\cos 3x$, simplifying LHS: $\cos(60^\circ+x)$. Nothing seems to work. Any hint?
2
votes
1answer
74 views

Product-to-sum trigonometry identity

I'm really not sure about this Product-to-sum identity on wiki. See: I cannot find this anywhere on the web - does anybody know a reference? Certainly the one wiki gives does not cover it. I'm ...
0
votes
3answers
50 views

Finding solution of a Trigonometric equation

$$\tan A + \tan 2A + \tan 3A=0$$ I tried converting these all in sin and cos and I got the answer but the answer didn't match. So any one could just help me. Thanks
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3answers
53 views

Solving Trigonometric Problems Like These

I am not sure how to approach this problem at all. I have no idea where to start or what it wants from me. Find the exact value of $\sin \theta$ and $\cos \theta$ given that $\cos ...
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0answers
55 views

Is my answer correct, or textbook correct?

My answer is 9.45, textbook is 4.7 I did cos 38 = x / 12, 12 cos 38 = x 9.45 = x ty in advance
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2answers
52 views

Converting $5\sin 60^\circ$ to $5\sqrt{3} / 2$.

$$\eqalign{\dfrac x5&=\sin60^\circ \\ x&=5\sin60^\circ\\&=\dfrac{5\sqrt{3}}{2}}$$ Can someone tell me how the last part was derived? How do I get from $5\sin 60^\circ$ to $5\sqrt{3} / 2$? ...
0
votes
1answer
90 views

How to find $\theta$ for $\tan\theta=-\frac{4}{3}$?

Given $\tan\theta = -\frac{4}{3}$, between $0\leq\theta\leq2\pi$, how can I find both values of $\theta$, with or without a calculator?
0
votes
1answer
65 views

Triangulation in delayed loudspeaker setup

I could use some help with the following situation. Two physically displaced speakers need to arrive on time, in order to achieve summation, for a given position (green dot) within a listening plane ...
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2answers
168 views

Integrate $\int \sin^4x \cos^2x dx$

Integrate $$\int \sin^4x \cos^2x dx$$ Now, there's few solutions to this problem already on the internet. For example on yahoo: https://answers.yahoo.com/question/index?qid=20090204203206AAbjUfM and ...
0
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2answers
167 views

Thinking of sohcahtoa with 90 in a triangle.

I know the answers from a unit circle. But when looking at a triangle how do you interpret Angle C sin C = cos C = tan C = I know the cos 90 = 0 and ...
0
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2answers
39 views

Trigonometry Identities questions

Given that $\sin\theta =\dfrac15$ and $0<\theta <\dfrac{\pi}2$, without evaluating the angle $\theta$, find the exact value of $$\sin\left( \frac{\theta}2-\theta \right)\tag1$$ I know that ...
3
votes
2answers
3k views

Trig Identity Proof of Tan(x) + Tan(y)

I am trying to prove the identity below to help with the simplification of another function that I'm investigating as it doesn't appear to be a standard trig identity. $$ \tan\left(x\right) + ...
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1answer
82 views

Trigonometry Question, Finding the distance and angle of elevation.

So there is this question, and for some reason, whether it be the early time of day or my lack of skills, It seems I have no idea how to draw the required diagram. I have tried and tried but none of ...
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0answers
71 views

Law of Sines Non-Euclidean geometry

Is the following Law of Sines valid on all surfaces isometric to a sphere? $$\frac{\sin A }{ \sin a }= \cdots = \frac{ \sin C }{ \sin c } = E.$$ And similarly, Is the following Law of hyperbolic ...
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2answers
156 views

Calculating appearance of size of object at given distance

Here's the problem I'd like to solve. If I'm 1 ft away from a computer screen and a word on the screen appears a certain size, is there an equation or calculation that will tell me how big that ...
0
votes
1answer
74 views

Find the value of $\sin\frac{19 π}{2}$ using the addition and subtraction trigonometry formulas

The formula is $\displaystyle\sin(s-t)= \sin(s)\cos(t) - \cos(s)\sin(t)$ $$\sin \frac{19 π}{2}=\sin \left(\frac{21 π}{2} - \frac{2π}{2}\right)$$ I am not sure I know how to convert radians. I would ...
0
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5answers
77 views

Solve $x$ for equation : $\sin^2(x) - \cos(x) - 1 = 0$

I am trying to solve $x$ for $\sin^2(x) - \cos(x) - 1 = 0$, for $0°\leqslant x \lt 360°$. I have the key with the answer $0°$ but have been unable to confirm this using Wolfram Alpha (I assume I ...
1
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1answer
71 views

Inverse Laplace transform, none factorable denominator

I am really stumpted on this problem and can't seem to figure out where to go from where I am. Can anyone give me some advice or hint where I should do next? Here is the problem: ...
2
votes
3answers
66 views

Prove this trig identity

$$\sin^2(x) - \cos^2(x) - \tan^2(x) = \frac{2\sin^2(x) - 2\sin^4(x) - 1 }{ 1-\sin^2(x)}.$$ I tried this but I can't figure out how they got $-2\sin^4(x)$.
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0answers
43 views

Worded trig question

From the top of a vertical cliff 125m high, the angle of depression to the top of a ship is 3.5°. If the ship is 1800m from the base of the cliff, determine the height of the ship above the water. I ...
2
votes
1answer
48 views

Parametric & Trigonometry

$$x=7\sin(t)+\sin(7t)$$ $$y=7\cos(t)+\cos(7t)$$ How would I solve this one out? I have to simplify the two enough to graph it. I squaring the two and adding them together, but I hit a roadblock: ...
3
votes
1answer
48 views

For which angles is inequality true

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 6: a) For which angles $\alpha$ is $\sin^4\alpha-\cos^4\alpha > \sin^2\alpha-\cos^2\alpha$ ? b) For which angles ...
26
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
0
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2answers
60 views

Trigonometric equation with generalized solution

The equation $$\cot A - \tan A=2$$ Find the generalized form of $A$ I got the answer and the answer came $$\tan A=(2^{1/2})-1 \text{ and } -2^{1/2}-1$$ Now how to write the generalized form j ...
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5answers
124 views

Verify algebraically that the equation $\frac{\cos(x)}{\sec(x)\sin(x)}=\csc(x)-\sin(x)$ is an identity

I am stuck when I get to this point $\frac{\cos^2(x)}{\sin(x)}$. Am I on the right track? Verify algebraically that the equation is an identity: ...
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3answers
7k views

Related Rates Ladder Problem with Angles

The problem is as follows: A 13-foot ladder leans against the side of a building, forming an angle θ with the ground. Given that the foot of the ladder is being pulled away from the building at the ...
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3answers
183 views

Evaluate $\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$

$$\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$$ My approaches so far has been using substitution with $\tan x = t$ and $\tan \frac x2 = t$ but the calculations has been harder than I ...
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0answers
93 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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4answers
111 views

Derivative with respect of a function

i have a function of two variables: $f(\theta,\phi) = \theta \sin(\phi)$ and i have to differentiate $f(\theta,\phi)$ with respect to: $1 - 0.5\theta^2$ That is: ...
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1answer
100 views

Solving right triangle given the area and one angle

Given right angle triangle $ACB$ (C is the right angle) has an area of 224 $mm^2$, what is the length of leg b if angle A equals 31.7deg? Here's the scenario: I have one right triangle completely ...
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1answer
123 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
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2answers
64 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
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2answers
50 views

Trig Question, Please help.

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $\cot \theta = −4$, $\sin \theta > 0$ Then it asks me to find: $ \sin(\theta) $, ...