 # High school trigonometry questions and answers

Recent questions in Trigonometry Sereinserenormg 2022-01-23 Answered

### The range of $f\left(x\right)={\mathrm{cos}}^{2n}\left(x\right)+{\mathrm{sin}}^{2n}\left(x\right)$ I have I need to find the range of the function. I took n=2 and I got $f\left(x\right)=1-{\frac{12}{\mathrm{sin}}}^{2}\left(2x\right)$ and the range of this is [1/2,1] Also, for n=3 I got [1/4,1]. How to find the range for n? Jaylene Franco 2022-01-23 Answered

### Finding solutions to $3{\mathrm{tan}}^{2}\theta -\mathrm{tan}\theta -14=0$ within $0\le \theta \le {360}^{\circ }$ Adrien Barron 2022-01-23 Answered

### Determine the greatest possible value of $\sum _{i=1}^{10}\mathrm{cos}3{x}_{i}$ for real numbers ${x}_{1},{x}_{2}\dots {x}_{10}$ satisfying $\sum _{i=0}^{10}{\mathrm{cos}x}_{i}=0$ My attempt: $\sum \mathrm{cos}3x=\sum 4{\mathrm{cos}}^{3}x-\sum 3\mathrm{cos}x=4\sum {\mathrm{cos}}^{3}x$ So now we have to maximize sum of cubes of ten numbers when their sum is zero and each lie in interval [−1,1]. i often use AM GM inequalities but here are 10 numbers and they are not even positive. Need help to how to visualize and approach these kinds of questions. Jamya Elliott 2022-01-23 Answered

### Solving $4{\mathrm{sin}}^{4}x-4{\mathrm{sin}}^{2}x+1=0,x\in \left[0,2\pi ⟩$ I am having problem solving this trig equation. Every time I try to simplify or substitute sinx with u, I get $x=\mathrm{\varnothing }$. How do I go about solving this equation? meteraiqn 2022-01-22 Answered

### How do you find the $\mathrm{arcsin}\left(\mathrm{sin}\left(\frac{7\pi }{6}\right)\right)$ ? Carly Shannon 2022-01-21 Answered

### What is the inradius of the octahedron with sidelength a? 1) Determine the height of one of two constituent square pyramids by considering a right triangle, using the fact that the height of the equilateral triangle is $a\sqrt{\frac{3}{2}}$. (Incidentally, by this you obtain the circumradius of the Octahedron.) 2) Now cut the (solid) octahedron along 4 of these latter heights, i.e. across two non-adjacent vertices and two midpoints of parallel sides of the ''square base''. (Figure 1) Consider the resultant rhombic polygon, whose sidelength is $a\sqrt{\frac{3}{2}}$. Choose a convenient pair of adjacent corners A and B, s.t. the angle in A is obtuse. Then the insphere touches AB, say in M. So the question becomes: Is there a still simpler proof?  Yahir Haas 2022-01-21 Answered

### How do you simplify ${\mathrm{tan}}^{2}4b-{\mathrm{sec}}^{2}4b$ iocasq4 2022-01-21 Answered

### What is $\mathrm{sin}\left(2\mathrm{arcsin}\left(\frac{3}{5}\right)\right)$ ? blitzbabeiy 2022-01-21 Answered

### How do you calculate $\mathrm{tan}\left(\mathrm{arccos}\left(\frac{5}{13}\right)\right)$ ? veceriraby 2022-01-21 Answered

### Prove in detail the inequality of this formula: Let x, y, z positive real number and $\mathrm{△}ABS$ a triangle. $\left[ABC\right]$ denotes the triangle area and a, b, c the sides of the triangle. The inequality below is true: ${a}^{2}x+{b}^{2}y+{c}^{2}z\ge 4\left[ABC\right]\sqrt{xy+xz+yz}$ Fallbasiss4 2022-01-21 Answered

### I have a non-right-angled isosceles triangle with two longer sides, X, and a short base Y. I know the length of the long sides, X. I also know the acute, vertex angle opposite the base Y, lets Stefan Hendricks 2022-01-17 Answered

### How to solve it without integrals: $\sum _{k=1}^{n}\mathrm{sin}\left(\frac{k\pi }{n}\right)\mathrm{sin}\left(\frac{k\pi }{{n}^{2}}\right)$ Francisca Rodden 2022-01-17 Answered

### Evaluate this integral ${\int }_{0}^{\frac{\pi }{4}}\frac{\mathrm{cos}2x}{\mathrm{cos}x+\mathrm{sin}x}dx$ Ernest Ryland 2022-01-17 Answered

### Prove that the series $\sum _{i=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n}}\mathrm{sin}\left(\frac{1}{\sqrt{n}}\right)$ is divergent petrusrexcs 2022-01-17 Answered

### A simple equation involving trigonometry If $\mathrm{cos}\theta +\mathrm{sin}\theta +\mathrm{tan}\theta =\sqrt{2}+1$ then what is the maximum value of $\mathrm{sin}\theta ,{\mathrm{sin}}^{2}\theta$? Kathy Williams 2022-01-17 Answered

### Integral of $sech\left(x\right)$ Cheexorgeny 2022-01-16 Answered

### What is the general solution to the equation $\mathrm{sin}x+\sqrt{3}\mathrm{cos}x=\sqrt{2}$ I need to find the general solution to the equation $\mathrm{sin}x+\sqrt{3}\mathrm{cos}x=\sqrt{2}$ So I went ahead and divided by 2, thus getting the form $\mathrm{cos}\left(x-\frac{\pi }{6}\right)=\mathrm{cos}\left(\frac{\pi }{4}\right)$ Thus the general solution to this would be $x=2n\pi ±\frac{\pi }{4}+\frac{\pi }{6}$ Which simplifies out to be, $x=2n\pi +\frac{5\pi }{12}$ $x=2n\pi -\frac{\pi }{12}$ PEEWSRIGWETRYqx 2022-01-16 Answered

### Proving $\frac{1}{{\left(\mathrm{tan}\left(\frac{x}{2}\right)+1\right)}^{2}{\mathrm{cos}}^{2}\left(\frac{x}{2}\right)}=\frac{1}{1+\mathrm{sin}x}$ zeotropojd 2022-01-16 Answered

### Is my evaluation of complex trigonometric expression $\mathrm{sin}\left(\frac{\pi }{4}+2i\right)-\mathrm{sin}\left(2i\right)$ correct? abreviatsjw 2022-01-16 Answered

### Why does $\mathrm{cot}\left(2\mathrm{arctan}\left(Ax\right)\right)=\frac{1-{\left(Ax\right)}^{2}}{2Ax}$

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