📏 trigonometry

1. **State the problem:** We want to find the value of $$\arcsin\left(\sin\left(\frac{100\pi}{11}\right)\right)$$. 2. **Recall the domain and range:** The function $$\arcsin(x)$$ r

1. **Problem statement:** We need to find the height $n$ of a flagpole given a right triangle where the angle between the wire (hypotenuse) and the ground is $72^\circ$, and the ho

1. **State the problem:** We need to find the height $n$ of a flag pole. The pole, the ground, and the wire form a right triangle. The angle between the wire and the ground is $72^

1. **State the problem:** A farmer wants to build a fence around a right-angled triangular field. One side adjacent to the right angle is 151 m, and the angle opposite this side is

1. The problem is to convert the polar coordinate $\left(7, \frac{3\pi}{4}\right)$ to Cartesian coordinates $(x,y)$. 2. The formulas to convert from polar to Cartesian coordinates

1. **Problem Statement:** Find an equation of the form $y = a \cos(bx)$ for the given graph. 2. **Identify amplitude $a$:** The amplitude is half the distance between the maximum a

1. **State the problem:** We need to find an equation of the form $y = a \sin(x)$ that matches the given sine wave graph. 2. **Analyze the graph:** The sine wave oscillates between

1. **State the problem:** We need to find the equations of the sine and cosine functions that match the given graph, expressed as translations of $y=\sin(x)$ and $y=\cos(x)$ with a

1. **Problem Statement:** We are given a cosine graph that is horizontally shifted. We need to find the equations for the closest left and right horizontal shifts of the form $$y =

1. **Problem Statement:** We are given a cosine graph that has been vertically shifted. The general form of the function is $$y = \cos(x) + C$$ where $$C$$ is the vertical shift.

1. **Problem statement:** We have a right-angled triangle with one angle $y$ to find. The side opposite angle $y$ is 7 cm, and the side adjacent to angle $y$ is 8 cm. 2. **Formula

1. **Problem Statement:** We have a right triangle TUS with a right angle at U, angle at S measuring 59°, and side TU opposite angle S with length 7 units. We need to find the leng

1. **State the problem:** Solve the equation $$9 \sin 2\theta - 28 \cos 2\theta = 90$$ for $$\theta$$. 2. **Recall the formula and approach:** The equation is a linear combination

1. **State the problem:** We are given a triangle ABC with angle ACB = 38°, side AB = 29 mm, and side AC = 40 mm. We need to find the two possible sizes of angle BAC to 3 significa

1. **Problem:** Solve the equation $\sin^2 x - 3 = 2 \sin x$. 2. **Rewrite the equation:** Move all terms to one side:

1. **State the problem:** Prove that $$\cos^2\theta + \tan\theta \csc\theta = \cos\theta$$. 2. **Recall the definitions and identities:**

1. **State the problem:** Simplify the expression $$\frac{1 - \cos^2 x}{1 + \sin^2 x}$$ and verify if it equals $$-\sin x$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \c

1. The problem is to solve the equation $$\frac{\cos 3x}{2} = 0$$ for $x$. 2. First, multiply both sides by 2 to eliminate the denominator:

1. **State the problem:** Given that $\sec \theta - \tan \theta = \frac{1}{4}$, find $\sec \theta + \tan \theta$. 2. **Recall the identity:** We know that $$(\sec \theta - \tan \th

1. **Problem statement:** Sketch the functions $f(x) = 2\sin x$ and $g(x) = \sin x + 1$ for $x \in (0^\circ, 360^\circ)$. Find the period of $f$ and the amplitude of $g$.

1. **State the problem:** Simplify the expression $$\frac{1+\sin^2 x}{\cos^2 x}$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^2 x = 1$$.