4
votes
4answers
55 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} ...
2
votes
0answers
50 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
27 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
36 views

Derivation of sine and cosine case

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
56 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
66 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
40 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
58 views

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

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

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

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
57 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
70 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
96 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
104 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
49 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 ...
0
votes
2answers
41 views

Express $\sin(x) + \sqrt{3}\cos(x)$ in the form $A\sin(x + a)$ [closed]

How would I go about expressing $\sin(x) + \sqrt{3}\cos(x)$ in the form $A\sin(x + a)$, where $A > 0$ and $0 < a < \pi/2$?
0
votes
3answers
48 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
4answers
56 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
27 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
109 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
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?
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 ...
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 ...
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 ...
6
votes
7answers
150 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 $\cos^2\alpha-\sin^2\alpha$. ...
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 ...
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 ...
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
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 ...
0
votes
1answer
38 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 ...
2
votes
1answer
38 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
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 ...
1
vote
5answers
39 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: ...
1
vote
3answers
31 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 ...
2
votes
3answers
64 views

Using the Chain Rule to prove trig derivatives

I'm having trouble with this problem, I'm not sure how to tackle it and I was wondering if somebody could set me on the right path. The problem is as follows: Use the Chain Rule to show that if ...
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. ...
1
vote
2answers
31 views

Simple Trigonometric Equation

I am asked to solve the trigonometric equation $2cos \theta = \sqrt 3$ I rearrange it to $cos\theta = \frac{\sqrt3}{2}$ Now, at this point I am not sure what to do? Can someone describe to me the ...
0
votes
1answer
65 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 ...
0
votes
1answer
17 views

Discovering the derivatives of functions combined with trig values.

Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose $m(\frac{\pi}{3}) = 4$ and $ m'(\frac{\pi}{3}) = -2$, and let $g(x) = m(x)\sin x$ ...
1
vote
2answers
49 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
40 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 ...
4
votes
2answers
46 views

If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5>\ldots?$

Problem : If $x,y \in (0,\frac{\pi}{2})$ then expression $\sin x +\cos y +\tan^2y+\cot^2x+5$ is always greater than : (a) $\ 7 $ (b) $\ 8 $ (c) $\ 9 $ (d) $\ $none of these Solution : We ...
2
votes
4answers
177 views

Solve The Triangle

I am having a tough time trying to solve this problem. I have utilized the 30, 60, 90 triangle measures for the length of sides. However, I am stuck since the side that would be √3 has 100 as its ...
1
vote
2answers
53 views

Finding the derivative of sinus and cosinus. Trigonometric identities

How can we see that $$\sin(x+h)-\sin(x)=2\sin\left(\frac h2\right)\cos\left(x+\frac h2\right)$$ How can we see that $$\cos(x+h)-\cos(x)=-2\sin\left(\frac h2\right)\sin\left(x+\frac h2\right)$$ Do ...
3
votes
2answers
47 views

Trigonometry sum of solutions question

Problem: For which $a$ will the sum of solutions be equal to $100$, in $\sin(\sqrt{ax-x^2})=0$. The attempt at a solution: For $\sin(x)=0$, $x$ must be equal to $0$, so we get ...
2
votes
1answer
34 views

Trigonometric identity proof problem

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 5: d) $\frac{\sin\alpha}{1+\cos\alpha}=\frac{1-\cos\alpha}{\sin\alpha}$ I would appreciate some hints on how to ...
1
vote
2answers
51 views

Find the exact value of the trigonometric function $\sin 7\pi/ 6$

I am finding it a little difficult to solve this problem. The reference angle for $\sin 7\pi/6$ is sin 30 degrees (I think) which is sine 1/2. But that is not the answer. How do I sove this problem?