 High school trigonometry questions and answers

Recent questions in Trigonometry Marla Payton 2021-12-28 Answered

Eliminate $$\displaystyle\theta\ {\quad\text{and}\quad}\ \phi$$ between the following equations: $$\displaystyle{\left\lbrace\begin{array}{c} {\sin{\theta}}+{\sin{\phi}}={x}\\{\cos{\theta}}+{\cos{\phi}}={y}\\{\tan{{\frac{{\theta}}{{{2}}}}}}{\tan{{\frac{{\phi}}{{{2}}}}}}={z}\end{array}\right.}$$ What I've done so far I've established that $$\displaystyle{\tan{{\left({\frac{{\theta+\phi}}{{{2}}}}\right)}}}={\frac{{{\sin{\theta}}+{\sin{\phi}}}}{{{\cos{\theta}}+{\cos{\phi}}}}}$$ so that $$\displaystyle{\tan{{\frac{{\theta+\phi}}{{{2}}}}}}={\frac{{x}}{{y}}}$$ I then used the trigonometric identity $$\displaystyle{\tan{{\left({\frac{{\theta+\phi}}{{{2}}}}\right)}}}={\frac{{{\tan{{\frac{{\theta}}{{{2}}}}}}+{\tan{{\frac{{\phi}}{{{2}}}}}}}}{{{1}-{\tan{{\frac{{\theta}}{{{2}}}}}}{\tan{{\frac{{\phi}}{{{2}}}}}}}}}$$ and with a little manipulation got to $$\displaystyle{\tan{{\frac{{\theta}}{{{2}}}}}}+{\tan{{\frac{{\phi}}{{{2}}}}}}={\frac{{{x}{\left({1}-{z}\right)}}}{{{y}}}}$$ I'm stumped on the next steps Anne Wacker 2021-12-28 Answered

Determining the period of $$\displaystyle{\frac{{{\sin{{2}}}{x}}}{{{\cos{{3}}}{x}}}}$$ I would like to compute the period of this function which is a fraction of two trigonometric functions. $$\displaystyle{\frac{{{\sin{{2}}}{x}}}{{{\cos{{3}}}{x}}}}$$ Is there a theorem for this? what trick to use to easily find the period? I started by reducing the fraction but I'm stuck on the rest. For example, let T be the period to be calculated: $$\displaystyle{\frac{{{\sin{{2}}}{x}}}{{{\cos{{3}}}{x}}}}={\frac{{{\sin{{\left({2}{x}+{2}{T}\right)}}}}}{{{\cos{{\left({3}{x}+{3}{T}\right)}}}}}}={\frac{{{\sin{{2}}}{x}{\cos{{2}}}{T}+{\sin{{2}}}{T}{\cos{{2}}}{x}}}{{{\cos{{3}}}{x}{\cos{{3}}}{T}-{\sin{{3}}}{T}{\sin{{3}}}{x}}}}$$ Gregory Jones 2021-12-28 Answered

Finding solution of a Trigonometric equation $$\displaystyle{\tan{{A}}}+{\tan{{2}}}{A}+{\tan{{3}}}{A}={0}$$ I tried converting these all in sin and cos and I got the answer but the answer didn't match Kathleen Rausch 2021-12-27 Answered

Why is $$\displaystyle{\tan{\theta}}\approx{\frac{{{1}}}{{{\frac{{\pi}}{{{2}}}}-\theta}}}$$ for $$\displaystyle\theta$$ close to $$\displaystyle{\frac{{\pi}}{{{2}}}}$$? I wanted to see what the behaviour of the steep part of the $$\displaystyle{\tan{}}$$ curve was like, i.e. the behaviour of $$\displaystyle{\tan{{\left({x}\right)}}}\ {a}{s}\ {x}\to{\left({\frac{{\pi}}{{{2}}}}\right)}^{{-{}}}$$. So by thinking about a shift of the graph of $$\displaystyle{\tan{{\left({x}\right)}}}\ {b}{y}{\frac{{\pi}}{{{2}}}}$$ to the left, I put some small (positive and negative) values of $$\displaystyle\theta$$ into my calculator for the function $$\displaystyle{\tan{{\left(\theta+{\frac{{\pi}}{{{2}}}}\right)}}}$$ $$\displaystyle{\tan{\theta}}\approx{\frac{{{1}}}{{{\frac{{\pi}}{{{2}}}}-\theta}}}$$ for $$\displaystyle\theta$$ close to $$\displaystyle{\frac{{\pi}}{{{2}}}}$$? or, in more colloquial terms, The steep part of $$\displaystyle{\tan{{x}}}$$ is just like the steep part of $$\displaystyle{\frac{{1}}{{x}}}$$ But why is this the case? I couldn't deduce it easily using the Maclaurin expansion of $$\displaystyle{\tan{{\left({x}\right)}}}$$. Is there a more intuitive explanation? I couldn't think of any explanations analogous to those explaining small angle approximations. Carla Murphy 2021-12-27

Evaluating $$\displaystyle\lim_{{{n}\to\infty}}{\left({1}+{\frac{{{\sin{{n}}}}}{{{5}{n}+{1}}}}\right)}^{{{2}{n}+{3}}}$$ (a $$\displaystyle{1}^{{\infty}}$$ indeterminate form) agreseza 2021-12-27 Answered

Prove that $$\displaystyle{\sin{{x}}}+{\cos{{x}}}=\sqrt{{2}}{\sin{{\left({x}+{\frac{{\pi}}{{{4}}}}\right)}}}$$ $$\displaystyle\sqrt{{2}}{\left({x}+{\frac{{\pi}}{{{4}}}}\right)}=\sqrt{{2}}{\left({\sin{{x}}}{\cos{{\frac{{\pi}}{{{4}}}}}}+{\cos{{x}}}{\sin{{\frac{{\pi}}{{{4}}}}}}\right)}={\sin{{x}}}+{\cos{{x}}}$$ Could you solve it from opposite? $$\displaystyle{\sin{{x}}}+{\cos{{x}}}=\sqrt{{2}}{\sin{{\left({x}+{\frac{{\pi}}{{{4}}}}\right)}}}$$ Dowqueuestbew1j 2021-12-27 Answered

Solve this trigonometric equation $$\displaystyle{\sin{{2}}}{x}-\sqrt{{3}}{\cos{{2}}}{x}={2}$$ I tried dividing both sides with $$\displaystyle{\cos{{2}}}{x}$$ but then I win $$\displaystyle{\frac{{{2}}}{{{\cos{{2}}}{x}}}}$$ Alan Smith 2021-12-27 Answered

How do you simplify $$\displaystyle{\sin{{\left({{\tan}^{{-{1}}}{\left({x}\right)}}\right)}}}$$? Salvatore Boone 2021-12-27 Answered

What is $$\displaystyle{\sin{{\left({x}\right)}}}+{\cos{{\left({x}\right)}}}$$ in terms of sine? Alan Smith 2021-12-26 Answered

$$\displaystyle{\arctan{{\left({\frac{{{x}+{1}}}{{{x}-{1}}}}\right)}}}$$ to power series I want to find an expression for $$\displaystyle{\arctan{{\left({\frac{{{x}+{1}}}{{{x}-{1}}}}\right)}}}$$ as a power series, with $$\displaystyle{x}_{{0}}={0}$$, for every $$\displaystyle{x}\ne{1}$$ My initial thought was to use the known $$\displaystyle{\arctan{{\left({x}\right)}}}={\sum_{{{n}={0}}}^{{\infty}}}{\frac{{{\left(-{1}\right)}^{{n}}{x}^{{{2}{n}+{1}}}}}{{{2}{n}+{1}}}}$$ but I don't know how to keep going if I replace x with $$\displaystyle{\frac{{{x}+{1}}}{{{x}-{1}}}}$$ William Boggs 2021-12-26 Answered

If $$\displaystyle{2}{\sin{\theta}}+{\cos{\theta}}=\sqrt{{3}}$$, what is the value of $$\displaystyle{{\tan}^{{2}}\theta}+{4}{\tan{\theta}}$$ ? Inyalan0 2021-12-26 Answered

What is the value of $$\displaystyle{\frac{{{1}}}{{{{\cos{{18}}}^{{\circ}}{\sin{{9}}}^{{\circ}}}}}}+{\frac{{{1}}}{{{{\cos{{18}}}^{{\circ}}{\cos{{9}}}^{{\circ}}}}}}$$? Irrerbthist6n 2021-12-26 Answered

Finding all a such that $$\displaystyle{f{{\left({x}\right)}}}={\sin{{2}}}{x}-{8}{\left({a}+{1}\right)}{\sin{{x}}}+{\left({4}{a}^{{2}}+{8}{a}-{14}\right)}{x}$$ is increasing and has no critical points Obviously, the first thing I did was to find the derivative of this function and simplify it a bit and I got: $$\displaystyle{f}'{\left({x}\right)}={4}{\left({{\cos}^{{2}}{x}}-{2}{\left({a}+{1}\right)}{\cos{{x}}}+{\left({a}^{{2}}+{2}{a}-{4}\right)}\right)}$$ But now how do I proceed further, had it been a simple quadratic in x. expeditiupc 2021-12-26 Answered

Find the general solution of the equation $$\displaystyle{\sin{{x}}}+{\sin{{2}}}{x}+{\sin{{3}}}{x}={0}$$. I have started doing this problem by applying the formula of $$\displaystyle{\sin{{A}}}+{\sin{{B}}}$$ but couldn't generalise it. Please solve it for me. hvacwk 2021-12-26

I have the following equation: $$\displaystyle{\sin{{\left({x}-{\frac{{\pi}}{{{6}}}}\right)}}}+{\cos{{\left({x}+{\frac{{\pi}}{{{4}}}}\right)}}}={0}$$ Terrie Lang 2021-12-26 Answered

Simplify $$\displaystyle{\sin{{\left({\frac{{\pi}}{{{3}}}}\right)}}}$$ kuvitia9f 2021-12-26 Answered

What is $$\displaystyle{{\cos}^{{-{1}}}{\left({\frac{{{1}}}{{{2}}}}\right)}}$$ ? Agohofidov6 2021-12-26 Answered

How do you find the terminal point on the unit circle determined by $$\displaystyle{t}={5}\frac{\pi}{{12}}$$ ? Margie Marx 2021-12-26 Answered

How do you calculate the slope, x intercept, and y intercept of the following equation: $$\displaystyle{f{{\left({x}\right)}}}=−{6}{x}−{3}$$? Judith McQueen 2021-12-26 Answered

What does $$\displaystyle{\cos{{x}}}{\sin{{x}}}$$ equal?

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