Solving Trigonometry Equations and Problems with Examples and Tips

Given that
$5 \cos (x) - 12 \sin (x) = 13$
Im

Aubrey Hendricks 2022-01-27

Why does $\cos θ = \sin (\frac{π}{2} - θ)$ even when $θ > 90^{\circ}$ ?
I get why this relationship holds true in 90 degree triangles, but I don't get why this relationship holds true when $θ > 90^{\circ}$ ? I get that when $θ > 90^{\circ}$ you find the acute angle that describes $θ$ , if you see what I mean. But there is no way of showing that in this function! So why would the fact that $\cos θ = \sin (\frac{π}{2} - θ)$ still hold true here?

Yahir Haas 2022-01-27

Solve $\tan (x - \frac{π}{4}) = - \tan (x + \frac{π}{2})$

Babenzgs 2022-01-27

General Sine and Cosine formula for sum of a finite number of angles
I was wondering is there is a general formula for $\sin (x_{1} + x_{2} + x_{3} + \dots + x_{n})$ as well as for the cosine function. I know that $\sin (x_{1} + x_{2}) = \sin (x_{1}) \cos (x_{2}) + \cos (x_{1}) \sin (x_{2})$ and $\cos (x_{1} + x_{2}) = \cos (x_{1}) \cos (x_{2}) - \sin (x_{1}) \sin (x_{2})$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern.

Advadlippabrj 2022-01-27

Find the least value for $\sin x - \cos^{2} x - 1$
At first I found the first derivative to be $y^{'} = \cos x + 2 \sin x \cos x$
Critical point $0, - \frac{π}{6}$ (principal)
$y - \sin x + 2 (\cos 2 x)$
Then substitution of x by critical points I found minima. But my answer is incorrect.
Correct minimum value is $- \frac{9}{4}$

Vilharjevw7 2022-01-27

Prove that $4 \cos^{4} x - 2 \cos 2 x - \frac{1}{2} \cos 4 x$ is independent of x
I don't know how to proceed. $\cos 4 x = 2 \cos^{2} (2 x) - 1$

kejpsy 2022-01-27

Want to do is find out how many solutions there are to the following equation in that interval (here A is a constant): $\cos (A \cos (\frac{s}{A})) = \sin (A \cos (\frac{s}{A})) = A \sin (\frac{s}{A}), s \in [0, 2 A π]$
I know it depends on A and I've tested a few cases, but I'm rusty on this trig stuff, so I haven't been able to do it yet

Gabriela Duarte 2022-01-27

Simplify $\sum_{k = 1}^{n} \tan (k) \tan (k - 1)$ by first proving $\tan (k) \tan (k - 1) = \frac{\tan (k) - \tan (k - 1)}{\tan (1)} - 1$

Ahmed Stewart 2022-01-27

Prove that $\cos (\frac{π z}{4}) - \sin (\frac{π z}{4}) = \prod_{n = 1}^{\infty} (1 + \frac{{(- 1)}^{n} z}{2 n - 1})$

Zugdichte2r 2022-01-27

What is the technique that should be used to evaluate $\int_{0}^{2 π} {(\sin (k \sin θ))}^{2} d θ$ ?

I searched for it and found out that it does

m4tx45w 2022-01-27

show that if $β$ is an acute angle and that $\tan β = 2 \sqrt{2}$ then $\cos β = \frac{1}{3}$
This question question has stumped me as I see no obvious way to go with it. This is all Ive

Gabriela Duarte 2022-01-27

Solving a differential equation with trigonometric functions
How could I approach
$y - A \cdot \sin (\frac{d^{2} y}{{d x}^{2}}) = 0$

Wendy Gutierrez 2022-01-27

Solving $\sin x = \sin y$ by using the prosthaphaeresis formulas
$\sin (x) - \sin (y) = 0$
$2 \cos (\frac{x + y}{2}), \sin (\frac{x - y}{2}) = 0$
There are two possibilities $\cos (\frac{x + y}{2}) = 0 or \sin (\frac{x - y}{2}) = 0$
In the first case:
$\frac{x + y}{2} = \frac{π}{2} + k π$
In the second case:
$\frac{x - y}{2} = k π$
The above two equations give the result:
$x = \frac{π}{2} + 2 k π$
$y = \frac{π}{2}$

jelentetvq 2022-01-27

How can I convert the complex number $i^{π}$ to trigonometric form?
I usually do these steps:
1. take Z=a+bi form,
2. find $r = \sqrt{a^{2} + b^{2}}$
3. $\cos (ϕ) = \frac{a}{r}, \sin (ϕ) = \frac{b}{r}$
4. find $ϕ$ from the above 2 equations.

Addison Gross 2022-01-27

Area bounded by $y = \sqrt{\frac{1 + \sin {x}}{\cos {x}}}$ and $y = \sqrt{\frac{1 - \sin {x}}{\cos {x}}}$

Miguel Davenport 2022-01-27

Let $x_{n} = 2^{n} a_{n}$ , and $a_{n + 1} = \sqrt{\frac{1 - \sqrt{1 - a_{n}^{2}}}{2}}, a_{0} = 1$ , how to prove $x_{n}$ converges?

Lainey Goodwin 2022-01-27

Easy way of tackling $\sqrt{x^{2}} + x = \sqrt{5}$
Solve for x

mairie0708zl 2022-01-27

If $\sin x + \sin^{2} x + \sin^{3} x = 1$ , then find $\cos^{6} x - 4 \cos^{4} x + 8 \cos^{2} x$

Octavio Miller 2022-01-27

Finding maximum and minimum values of $\sin (2 x) - x for x \in [- \frac{π}{2}, \frac{π}{2}]$

Kaydence Huff 2022-01-27

Approximating $\cos (47^{\circ})$
Given that $\cos (45^{\circ}) = \frac{\sqrt{2}}{2}$ what would $\cos (47^{\circ})$ be.
Using differential approximation, I get $\cos (47^{\circ})$ is about $\cos (\frac{45 π}{180}) - 2 \sin (\frac{47 π}{180}) = - 0.755600622$ which is of course not right as $\cos (47^{\circ}) = 0.68199836$

The majority of Trigonometry Math problems are quite easy to solve when you have the list of answers that help you see the logic and reasoning. Looking at various examples will help you to see how certain questions are related to provided solutions.

Remember that Trigonometry requires patience and analysis. By following your analysis skills, you will help yourself find the answers. If something remains unclear, start with your exploration of examples again and go step-by-step.

Precalculus

Practice Exercises for Trigonometry