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

learn more… | top users | synonyms (1)

6
votes
2answers
191 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
1
vote
1answer
36 views

Trigonometric inequality in a triangle

If $\alpha,\beta,\gamma$ are the interior angles in a triangle, the following inequality seems to hold: ...
0
votes
3answers
52 views

Simplify $\tan(360 - \theta)$

I am aware that $\tan(\alpha-\beta)=\dfrac{\tan(\alpha)-\tan(\beta)}{1+\tan(\alpha)\tan(\beta)}$ So for my question: $\tan(360 - \theta)$ Do I choose random value for $\theta$ and plug it into the ...
0
votes
3answers
58 views

Simplify $\sin (90 - \theta)$

Title. I have no idea what to do. Is their an identity I have to remember? What am I supposed to do to the equation? Do I have to solve for something first, what does it mean by simplify?
2
votes
2answers
56 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
0
votes
3answers
43 views

Trigonometric Identities help

How do you solve this? I can't figure out what I should do. $$\sin ^4\left(A\right)+\cos ^2\left(A\right)=\cos ^4\left(A\right)+\sin ^2\left(A\right)$$ Also, why is this equal zero? Can someone ...
12
votes
3answers
160 views

How to evaluate $\sum_{n=1}^{38}\sin\left(\frac{n^8\pi}{38}\right)$

Evaluate $$\sum_{n=1}^{38}\sin\left(\frac{n^8\pi}{38}\right)$$ I have found the problem on this page. I have no idea how to do it. Thank you very much.
2
votes
0answers
30 views

Solving systems of equations with trigonometric terms

I am trying to solve (or rather find the least squares solution for) a system of equations with trigonometric terms in them. The system consists of pairs of equations of the form $a_1 \cos\theta - ...
1
vote
2answers
32 views

The bird pointer problem: finding the angle of rotation

Suppose we have a bird pointer. He is a guy that likes to point at birds in the sky: His legs cannot move, however he can rotate around his torso. Also, his body and his arm always make a right ...
2
votes
2answers
77 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
4
votes
1answer
25 views

Family of lines $\sin\alpha x +\sin\beta y +\sin\gamma =0$

Problem : If $\sin(\alpha + \beta)\sin(\alpha -\beta) =\sin\gamma (2\sin\beta +\sin\gamma), 0 < \alpha , \beta ,\gamma <\pi$ then the family of lines $\sin\alpha x +\sin\beta y +\sin\gamma =0$ ...
2
votes
2answers
34 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: $$f(x)=\cos(x/3)+\cos(x/4).$$
0
votes
0answers
29 views

Trigonometry, find distance of arc movement

Imagine I have the setup as follows: I want to compute the height x in State 2, depending on how much the blue axis have moved. That is, the distance ...
2
votes
1answer
49 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
votes
0answers
15 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 ...
-3
votes
2answers
42 views

find a formula for $\sin3 \theta$ in terms of $\sin \theta$,$\cos \theta$ by using an angle-sum formula for sin(2θ+θ) [closed]

Find a formula for $\sin3 \theta$ in terms of $\sin \theta$,$\cos \theta$ by using an angle-sum formula for $\sin(2 \theta+\theta$)
2
votes
2answers
95 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, ...
-1
votes
1answer
25 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 ...
0
votes
1answer
22 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 ...
6
votes
7answers
151 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 $\cos^2\alpha-\sin^2\alpha$. ...
0
votes
1answer
38 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 ...
1
vote
0answers
32 views

Websites for math tests/quizzes

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
57 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
55 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) = ...
1
vote
2answers
40 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
votes
5answers
74 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
57 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
140 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 ...
1
vote
2answers
40 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
41 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
192 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
51 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
38 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
0
votes
3answers
44 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 ...
1
vote
0answers
45 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
0
votes
2answers
41 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
71 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
33 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 ...
0
votes
2answers
67 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 ...
1
vote
2answers
31 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
votes
2answers
24 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 ...
2
votes
2answers
241 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) + ...
1
vote
1answer
17 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 ...
1
vote
2answers
27 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
39 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
votes
5answers
70 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
vote
1answer
13 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: ...
1
vote
3answers
54 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 $-2\sin^4(x)$.
0
votes
0answers
35 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
39 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: ...