0
votes
1answer
35 views

Trigonometry Question - Tough one [on hold]

If in triangle ABC, sin A sin B sin C + cos A cos B = 1. Then find the value of sin C.
3
votes
3answers
43 views

Trigonometry Question: find Value of…

Find value of $3 + \cos2x + \cos4x + \cos6x - 4\cos x\cos2x\cos3x$. I tried with $\cos A + \cos B$ identity but it was not simplifying.... Help..
1
vote
4answers
53 views

Trigonometric functions of the acute angle

Find the other five trigonometric functions of the acute angle A, given that: \begin{align} &\text{a)}\ \ \sec A = 2 \\[15pt] &\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2} \end{align} ...
3
votes
2answers
53 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
3answers
138 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 ...
3
votes
1answer
186 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?
0
votes
1answer
68 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
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 ...
1
vote
1answer
15 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 ...
3
votes
1answer
45 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 ...
2
votes
1answer
28 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
0
votes
2answers
27 views

Applying the cosine even identity to the cosine difference identity

I'm slightly confused over what happens when you're applying cosine's "even identities" to the difference identity. Here's how I go about, please tell correct me as I feel i'm going wrong somewhere. ...
0
votes
1answer
41 views

Calculate length of radial intersecting a rectangle

In a rectangle like below, I need to calculate the length of any radial, from the center of the rectangle to where it intersects with the edge of the rectangle. Further, the angle of the radial is ...
1
vote
1answer
92 views

How Can I figure out when cosine = $\frac{2}{\pi}$?

So I'm doing Mean Value theorem homework which states $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ So I am trying to find $c$ for $f(x)=\sin x$ over the interval $[0,\frac{\pi}{2}]$. So using the Mean Value ...
0
votes
1answer
62 views

A few questions regarding the cosine difference identity

I've a few questions that stem from the proof given in my textbook regarding the cosine difference identity. The proof goes like this: Let $\alpha$ and $\beta$ be angles plotted in standard ...
-1
votes
1answer
48 views

Inverse Trigonometry proof

Please help me prove this equation as ive been trying for days and not able to solve the $\tan^{-1}( \cot^3 x)$ part. $$\tan^{-1}(\cot x)+\tan^{-1}(\cot^3 x)+\tan^{-1}(\frac{1}{2} \tan 2x)=0$$
0
votes
1answer
28 views

What is the effect of taking the sine of inverse cosine?

How can I evaluate the sine of an inverse cosine? for example: sin(arccos((x)^1/2))
1
vote
4answers
41 views

Finding the $\cot\left(\sin^{-1}\left(-\frac12\right)\right)$

How can I calculate this value? $$\cot\left(\sin^{-1}\left(-\frac12\right)\right)$$
1
vote
2answers
48 views

How to express a trigonometic equation in $\sin 2\theta $ and $\cos 2\theta $?

How do I express the given equation in $\sin 2\theta $ and $\cos 2\theta $ in terms of x? $x + 3 = 7\sin \theta $ with $\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $ for $\sin 2\theta ...
0
votes
2answers
39 views

Solving equations with powers without logarithms

Im taking an introduction to logarithms. Of course a short review of exponentiation is inherent for a clear understanding of logarithms. I was asked to find, for example, $27^x = 3$. (without the use ...
1
vote
1answer
26 views

Simple Trig Question / Introduction to Vectors Question

Sorry this is such a simple question; I'm just struggling a little with my trigonometry homework. An example question: "A ship sails due north (relative to the current) with a speed of 20 knots. The ...
0
votes
5answers
58 views

Taking the sin of arccos

When solving for the value of x in the equation $$\sin^{-1}{(\sqrt{2x})}=\cos^{-1}(\sqrt{x})$$ one would take the sin of both sides of the equation cancelling out the arcsin leaving ...
6
votes
2answers
525 views

Sum of this series

$$ \mbox{How do I find the sum of this series}\quad \sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}\ {\large ?} $$ Hints in the right direction would be appreciated.
2
votes
3answers
114 views

Evaluating a limit involving trigonometry

I really thank you for your answers to my first question--I could easily solve first problem and a few more ones without another question. But a while later I got another one while studying, then I ...
3
votes
4answers
133 views

How to solve: $\cos^2x + \sin x = 1$

$\cos^2x + \sin x = 1$ How to solve for $x$?
1
vote
3answers
25 views

Finding a point in a parallelogram

QUESTION: Find the point$(x,y)$ so that $(x,y)$ is in the first quadrant and $(x,y),(1,2),(4,10)$ and $(2,6)$ are vertices of a parallelogram.. I find this question very difficult.. Thanks...
0
votes
2answers
27 views

Express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$ using Euler's identity

Using Euler's identity ($e^{in\theta}=\cos n\theta+i \sin n\theta$), express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$. Any ideas?
2
votes
1answer
35 views

Limit of a Rational Trigonometric Function

When solving a trigonometric limit such as: $$\lim_{x \to 0} \frac{\sin(5x)}{\sin(4x)}$$ we rework the equation to an equivalent for to fit the limit of sine "rule": $$\lim_{x \to ...
2
votes
2answers
66 views

On proving an identity given a system of trig equations

We are given the following: $$a^2 + b^2 + 2ab\cos\theta = 1 \tag1$$ $$d^2 + c^2 + 2cd\cos\theta = 1 \tag2$$ $$ac + bd + (ad + bc)\cos\theta = 0\tag3$$ It is required to prove that: $$a^2 + c^2 = ...
3
votes
9answers
167 views

Find the exact value of $\sin (\theta)$ and $\cos (\theta)$ when $\tan (\theta)=\frac{12}{5}$

So I've been asked to find $\sin(\theta)$ and $\cos(\theta)$ when $\tan(\theta)=\cfrac{12}{5}$; my question is if $\tan (\theta)=\cfrac{\sin (\theta) }{\cos (\theta)}$ does this mean that because ...
2
votes
3answers
67 views

Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
1
vote
5answers
53 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
1
vote
2answers
37 views

How to find length of the sides of a triangle given the ratio of the sines of the sides?

Consider $\triangle ABC$. Let $\dfrac{\sin A}{\sin B} = \dfrac56$ and $\dfrac{\sin B}{\sin C} = \dfrac45$. Find $\dfrac{\vert AC\vert\cdot \vert AB\vert}{\vert BC\vert}$. If there is no definite ...
0
votes
1answer
34 views

$f(x)=sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
0
votes
3answers
83 views

Calculate $\frac{2\cos40^\circ-\cos20^\circ}{\sin20^\circ}$

I am trying to solve this task i.e. calculate this expression without using calculator, in terms of known values for angles such as 30,60,90,180 degrees :). ...
0
votes
1answer
34 views

Finding the range and domain of $h(x) = \sec (x)$

I am attempting to show how to find the range and domain of $h(x) = \sec (x)$. Here's my working so far. Consider $h(x) = \sec (x)$, which is defined as $h(x) = \sec (x)=\frac{1}{\cos(x)}$. We know ...
0
votes
1answer
60 views

Value of $\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$

I just put the values from the trignometric table to solve, but the answer is different in the answer book. $$\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$$
1
vote
2answers
35 views

Substitution of an implicit variable

I wasn't sure how to title this question: I want to manipulate the integral $$I(a,b) = \int_0^{\frac{\pi}{2}} \frac{d \phi}{\sqrt{a^2\cos^2 \phi + b^2 \sin^2 \phi}}$$ with this subsitution: $$\sin ...
1
vote
1answer
32 views

Trigonometry / Obtuse angle

If $\cos A = 4/5$ and $\sin B = 5/13$, where $A$ is a acute and $B$ is obtuse, find, without evaluating the angles $A$ and $B$, the values of a) $\sin (A-B)$ b) $\cos (A+B)$ I'm stuck figuring out ...
1
vote
2answers
32 views

Trigonometry / Finding the exact value

Given that $\cos \theta = \dfrac{-4}{5}$ and $\sin \theta$ is positive, obtain the exact values of $\cos (6\pi+\theta)$ i don't understand this question.
0
votes
2answers
34 views

Trigonometry reference angle of radian

Given that $-2\pi≤\theta≤0$ and $\theta$ has a reference angle of $\cfrac{\pi}{6}$ , find $\theta$ if it is in the a) 1st quadrant b) 2nd quadrant c) 3rd quadrant d) 4th quadrant I need help on ...
0
votes
2answers
38 views

Converting angles in Radians to degrees

convert following angles to degrees. Give your answer correct to 2 decimal points. a) $-3.5$ $-3.5 \times \dfrac{180}{π} = ?$ i'm stuck on this stage..
0
votes
3answers
38 views

Prove this trigonometric identity…

$$\sin({495}^{\circ})-\sin({795}^{\circ})+sin({1095}^{\circ})=0$$ So I have to prove that the identity is correct. How can I transform those large angles in smaller ones?
1
vote
2answers
33 views

Trigonometric ratios involving negative angles

Given that $\cos \theta = \dfrac{3}{5}$ and csc is positive a) which quadrant is $\theta$ in? Hence deduce the quadrant that $-\theta$ is in. so for question it is 2nd quadrant b) Without finding ...
1
vote
1answer
34 views

Providing solutions for the intersection of two trigonometric functions

I'm trying to find a general solution for the intersection of two trigonometric functions: $$a(x)=500\sin \left( \frac{\pi x}{2} \right)+150$$ $$a(x)=200\cos \left( \frac{\pi }{2}\left( x+\frac{\pi ...
2
votes
2answers
50 views

Finding all the values of $\theta$ for which $\tan(\theta)=\sqrt3$; problem with understanding.

My textbook has a section where it defines $\tan(\theta)$ as the following: "For acute angles $\theta$, $\tan(\theta)$ is the $y$-coordinate of the point on the terminal side of $\theta$ which lies on ...
2
votes
4answers
87 views

integration by parts of trig functions

Can anyone help me with this integral? $\int{x^3 \sin(x^4) dx}$ I set $u=x^3$, and I let $v=-\cos(x^4)$, so that $\frac{dv}{dx}=\sin(x^4)$ I tried using integration by parts, but, whenever I come ...
-3
votes
2answers
43 views

Trigonometric Identities involving sin cos, and other variables [closed]

If $$m\tan(X - 30) = n \tan(X + 120)\ ,$$ then find $\cos 2X$ in terms of $m$ and $n$.
0
votes
1answer
28 views

A question about the definition of the circular function $\tan(\theta)$

The circular function $\tan(\theta)$ is defined as $\tan (\theta)=\frac{\sin (\theta)}{\cos (\theta)}$. If we look at this in the context of the Unit Circle: From this picture it can be seen that ...