0
votes
1answer
31 views

What trig. identity would help solve $2 + \cos(2x) = 3\cos(x)$?

I need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = ...
3
votes
3answers
99 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
-2
votes
1answer
41 views

Problem involving Basic Trigonometry [on hold]

If $\sqrt{2} \cos{A}=\cos{B} +\cos^3{B}$, $\sqrt{2} \sin{A}=\sin{B} -\sin^3{B}$, then find the value of $\sin(A-B)$.
1
vote
3answers
29 views

Calculating the angle for a path between two nodes in a graph

I want to (programatically) draw an edge between two nodes in a graph, starting on the outside of the nodes. Below is an illustration of what I'm (poorly) trying to describe: I have the $(x,y)$ ...
3
votes
3answers
37 views

$x$-intercept of cosine graph

I am having problems understanding how to find the $x$-intercept of a cosine graph. Example: $10\cos(x/2)$ Answer:$((2n + 1)\pi , 0 )$ I have the answer just need help understanding the steps, ...
0
votes
1answer
33 views

How to find limits involving trigonometric functions as $x\to 0$?

Problem: find the limit as $x\rightarrow 0$ of $\dfrac{\tan(3x)}{\sin(2x)}$ $\dfrac{(\sin(2x) + 3)}{(\cos(7x)-8)}$ Note I am able to solve the first one using l'Hopitals, but I really want to be ...
0
votes
1answer
42 views

Find distance, given angles of elevation

Write an equation giving the distance d between the plane and observation post in terms of $\theta$ and $\phi$. Is this correct? when using the Law of Sines answer: $a/\sin\theta = c/\sin C$ ...
0
votes
3answers
71 views

How do I solve the trigonometric equation $\sec^3x - 2 \tan^2 x = 2$? [closed]

A friend asked to me how could she resolve this equation, but I don't know how to resolve it?? Could you help me?. The equation is : $\sec^3x - 2 \tan^2 x = 2$ Note: She told me that I can use ...
2
votes
1answer
73 views

What is the distance from the boat to the shoreline? [closed]

A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to the lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see ...
0
votes
4answers
142 views

Manipulating a trigonometric equation involving $\tan^2(3\theta)$ [closed]

If $\tan^23\theta = 1$, how do I manipulate the equation so I can make $\tan\theta$ the subject? I forgot how to do these since it has been a long time. I tried searching before posting. My answer is ...
1
vote
3answers
42 views

Bearings question

Kim leaves his house and walks for $2$ km on a bearing of $155^\circ$. How far south is Kim from his house now, to $1$ decimal place? I don't know where to start at all, the correct answer is ...
2
votes
1answer
34 views

Trigonometric Identity problem involving cot

Simplify $\displaystyle\frac{\cot25 + \tan65}{\cot25}$ My attempt is: $$\frac{\cot25 + \tan65}{\cot25}=\frac{\cot25 + \cot(90 - 65)}{\cot25}=\frac{\cot25 + \cot25}{\cot25}= \frac{\cot50}{\cot25}$$ ...
7
votes
2answers
171 views

Prove $\tan 54^\circ=\frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$

How to prove this identity without using the actual values of $\tan54^\circ$ and $\sin24^\circ$ $$\tan 54^\circ=\dfrac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$$ Edit: I still don't get it, I am ...
0
votes
1answer
26 views

Find the exact values without a calculator: (a) $\tan \frac{11\pi }6$ (b) $\sec \frac{-3\pi}4$ (c) $\cot \frac{-5 \pi}3$

Okay I know the unit circle back and forth, but I get confused when I am asked to find answers that do not refer to sine and cosine. For example, I am ask to evaluate $\tan \frac{11\pi }6$. Since ...
1
vote
3answers
24 views

Understanding trigonometric identities

Can someone help me understand trigonometric identities? For example, it is known that $\cos(90-\theta)$ is equal to $\sin \theta$, and vice versa. But why? Is it something to do with the unit circle? ...
0
votes
2answers
47 views

finding the value of $Z+Z^{2}+Z^{3}… $ if…

If $ Z+Z^{-1} = 2 \cos 5$ then what's the value of $Z+Z^{2}+Z^{3}.... ......Z^{63}$. I wanted to to solve this with the value of $Z$. But may be the value of $Z$ is complex. Now it's quite impossible ...
0
votes
3answers
40 views

Trigonometric Identities involving fractions

The question is to simplify: However, when I do that I end up with: $\frac{\cos\theta}{\frac{1}{\cos\theta}}$ Now, I don't know how to deal with these types of fractions. I have not encountered ...
1
vote
4answers
67 views

What does "sin$\theta > 0$ mean here?

The question is: If $\tan$ $\theta$ = -$\frac{8}{15}$, and $\sin$ $\theta$ > $0$, find $\cos$ $\theta$. What I did was draw a triangle on the unit circle with sides 8, 15 and therefore ...
-4
votes
1answer
29 views

Trigonometric Substitution LHS and RHS

Show by substitution into LHS and RHS that each trigonometric identity is satisfied by the given values of the angles. $(a)$ Show that $\sin2\theta=2\sin\theta\cos\theta$, when $\theta=150^\circ$. ...
0
votes
2answers
37 views

Exact value of a trigonometric ratio

I was asked to find the exact value of $\tan 240^\circ$. On my calculator, I type $\tan 240^\circ$, and then square the value to get a final answer of $\sqrt3$. However, the textbook answer says the ...
1
vote
2answers
26 views

Simplifying difference trig expression

Rewrite the following expression as a simplified expression containing one term: $$\cos (\frac{\pi}{3}+\varphi) \cos (\frac{\pi}{3}-\varphi) - \sin (\frac{\pi}{3}+\varphi) \sin ...
3
votes
1answer
48 views

How to solve the trigonometric equation $\cos (\pi\theta/\beta) - \cos(2\pi\theta/\beta)=0$?

I have a question regarding a problem I've been attempting to solve. It is an acceleration equation: $$a = ...
3
votes
1answer
27 views

Finding exact values of trig functions

Find exact value of each trigonometric function of $\theta$ if $\tan\theta=-1/5$ and $\sec \theta >0$ I know that $\cot \theta=-5,$ right? Secant and cosine are positive in the fourth ...
0
votes
4answers
48 views

Rewriting trigonometric expression in terms of $\cot x$

Rewrite the following expression in terms of $\cot x$: $$\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x}$$ I usually show my work on this site but I'm really lost about this problem. Any help ...
1
vote
2answers
27 views

Using sketch to find exact value of trigonometric expression

Use sketch to find exact value of $\tan (\cos^{-1}\dfrac{5}{13})$ I drew a right triangle with angle $\theta$ and sides $12,5,3.$ If $\cos \theta=\frac{5}{13},$ then $\sin \theta = \frac{12}{13}$ ...
2
votes
1answer
55 views

Help needed in verifying a trigonometric identity

I have the following identity: $$32\sin^{2}\left(\theta\right)\cos^{4}\left(\theta\right) =2 + \cos\left(2\theta\right) - 2\cos\left(4\theta\right) -\cos\left(6\theta\right) $$ I've tried ...
1
vote
2answers
26 views

Rewriting a trigonometric inequality (including a parameter)

How is it possible to rewrite these equations? $\sin{x}- \cos{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ≤ \frac{1 + \mu}{1 - \mu}$ and $\cos{x}- \sin{x} ≤ \mu(\cos{x} + \sin{x}) \implies \tan{}x ...
4
votes
4answers
65 views

How does $x^3 - \sin^3 x$ become $x^3 + \frac{1}{4}\sin{3x}-\frac{3}{4}\sin x$?

I was going through answers on this question and came across this answer and I was wondering how the user arrived at the first line where they state: $$f(x) \equiv x^3 - \sin^3 x = x^3 + {1 \over 4} ...
0
votes
1answer
53 views

Find $\sin(x+y)$, given $\tan x$ and $\cos y$

Given that $\tan x= -2$ and $\cos y= 1/2$ where $x$ and $y$ are in the 4th and 1st quadrants respectively. Find, without evaluating angles $x$ and $y$, a) $\sin (x+y)$ Here is what i have done so ...
2
votes
0answers
56 views

Eliminate variable in trigonometry equations

Say you have the equations: \begin{align} -S_1\sin\left(2\psi+\theta\right)+S_2\cos\left(\psi\right)&=S_3\\ S_1\cos\left(2\psi+\theta\right)+S_2\sin\left(\psi\right)&=S_4 \end{align} or ...
2
votes
3answers
34 views

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$?

Why $ (1- \sin \alpha + \cos \alpha)^2 = 2 (1 - \sin \alpha)(1+ \cos \alpha)$? I am learning trigonometric identities one identity I have to proof is the next: $ (1- \sin \alpha + \cos \alpha)^2 = ...
0
votes
3answers
46 views

Identity involving cosine of double angle

I am struggling to see this. I know that we can factor out $ a$, but I don't see how we can end up with the right hand side. $$a \cos ^2(a t)-a \sin ^2(a t)=a \cos (2 a t)$$
2
votes
4answers
61 views

Show that there is an angle $\theta$ such that $a=\cos\theta$ and $b=\sin\theta$

My problem is from Israel Gelfand's Trigonometry textbook. Page 50. Exercise 3: Suppose that $\alpha$ is some angle. If $a=4\cos^3\alpha-3\cos\alpha$ and $b=3\sin\alpha-4\sin^3\alpha$, show that ...
-2
votes
1answer
74 views

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$

For all $x$ in $[0,90]$ show that $\cos(\sin x ) >\sin(\cos x )$ I understood the solution given in my book which said  $$\cos(x)+\sin(x)\leq\sqrt{2}<90$$ $$\cos(x)<90-\sin(x)$$ But if ...
0
votes
3answers
44 views

Help needed verifying a trigonometric identity

I have the following identity: $$ \frac{\tan (t + h) - \tan(t)}{h} = \left( \frac{\tan (h)}{h} \right)\left( \frac{\sec^2(t)}{1 - \tan (t)\tan (h)} \right)$$ Having tried various approaches, ...
-2
votes
2answers
64 views

Prove that $16 \cos 12^\circ ·\cos 24^\circ ·\cos 48^\circ· \cos 96^\circ ·\cos 192^\circ = 1$ [closed]

Prove that $$16 \cos 12^\circ ·\cos 24^\circ ·\cos 48^\circ· \cos 96^\circ ·\cos 192^\circ = 1$$ Thanks.
2
votes
2answers
107 views

Prove that $\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18$ [closed]

Prove that $$\sin(12^\circ)\sin(48^\circ)\sin(54^\circ)=\frac18.$$ Without using a calculator. I tried all identities I know but I have no idea how to proceed. I always get stuck on finding ...
1
vote
1answer
42 views

Finding the zeros of trionometric polynomails.

I have a question about something I've struggled with for a while: Finding the zeros of trigonmetric polynomials. Let me show you a problem I am solving and you guys can tell me if I got the right ...
1
vote
5answers
63 views

Prove $\frac{\sin A}{\sec A+\tan A-1}+ \frac{\cos A}{\csc A+\cot A-1}=1$

$$\frac{\sin A}{\sec A+\tan A-1}+ \frac{\cos A}{\csc A+\cot A-1}=1$$ Prove that L.H.S.$=$R.H.S. This type of questions always creates problem when in right hand side some trigonometry function is ...
1
vote
4answers
77 views

inverse trigonometric equation $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$

I have problem with showing that $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$ I think there have to be used formula: $\displaystyle ...
2
votes
4answers
105 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
2
votes
9answers
111 views

Find $\tan x $ if $\sin x+\cos x=\frac12$

It is given that $0 < x < 180^\circ$ and $\sin x+\cos x=\frac12$, Find $\tan x $. I tried all identities I know but I have no idea how to proceed. Any help would be appreciated.
2
votes
4answers
64 views

Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$

Solve for $x$, $\tan x +\sec x = 2\cos x$ ; $−∞ < x < ∞$ $$\tan x + \sec x = 2\cos x$$ I tried changing it all to sin and cos $$\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2\cos x$$ then I ...
1
vote
4answers
67 views

How to memorize the trigonometric identities?

I am stuck trying to memorize the trig identities, and try as I may, I just can't get them to stick (especially the sum-product and product-sum formulas). I am concerned I won't be able to memorize ...
1
vote
5answers
67 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} ...
0
votes
1answer
28 views

Acute angle and trigonometric functions

Given that $\theta$ is an acute angle and $\cos\theta = \dfrac{7}{25}$. Find: $\tan\theta$, $\sin\theta$, $\sec\theta$.
0
votes
3answers
112 views

Simplify tan$\theta$ cos$\theta$

How do I simplify tan$\theta$ cos$\theta$ ? Why is this so hard to do? What pieces of information should I know before doing these? Can someone just tell me were am I going wrong? I have 5 days ...
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
60 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?
0
votes
3answers
45 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 ...