📏 trigonometry

1. The problem asks whether $\cos(87\pi + a)$ is positive or negative. 2. Recall the cosine addition formula and periodicity: cosine has period $2\pi$, so

1. We start with the equation: $$2\cos(2x) + 2\sin(2x) \cdot \cos(x) - 5\sin(x) \cdot \cos(2x) = 0$$ 2. Use double-angle identities: $$\cos(2x) = 2\cos^2(x) - 1$$ and $$\sin(2x) =

1. The problem is to understand the function $\sin x$ and its properties. 2. The sine function $\sin x$ is a periodic function with period $2\pi$, meaning $\sin(x + 2\pi) = \sin x$

1. **State the problem:** We are given the height function of a person on a Ferris wheel as $$h(t) = 20 \sin\left(\frac{\pi}{30}t - \frac{\pi}{2}\right) + 22$$ where $t$ is time in

1. **State the problem:** Solve the equation $$\tan \frac{1}{2}x = 3$$ for $$0 < x < 4\pi$$, giving answers in radians to 3 significant figures. 2. **Rewrite the equation:** Let $$

1. The problem states that angle $\theta$ is in quadrant IV and $\cos \theta = \frac{4}{5}$. We need to find $\tan \theta$. 2. Recall that in quadrant IV, cosine is positive and si

1. Problem: If angle $\theta$ is in quadrant IV and $\cos \theta = \frac{4}{5}$, find $\tan \theta$. 2. Since $\cos \theta = \frac{4}{5}$ and $\theta$ is in quadrant IV, $\sin \the

1. We are asked to verify the trigonometric identity: $$\sin 3A = \sin A (3 \cos^2 A - \sin^2 A)$$. 2. Start with the left-hand side (LHS): $$\sin 3A$$.

1. **State the problem:** Prove that $$\frac{1-\sin A}{1+\sin A} = (\sec A - \tan A)^2$$. 2. **Start with the right-hand side (RHS):**

1. Given that $\csc x = \frac{17}{15}$ and $x$ is a positive acute angle, we need to find the value of $x$. 2. Recall that $\csc x = \frac{1}{\sin x}$, so we have:

1. **State the problem:** Given $\tan x = -\frac{3}{4}$, find $\sin x$ and $\csc x$ for $x$ in the interval $0^\circ$ to $360^\circ$. 2. **Recall the identity:** $\tan x = \frac{\s

1. Given \( \tan x = \frac{3}{4} \), we need to find \( \sin x \) and \( \csc x \) for \( x \) in the interval \( 0^\circ \) to \( 360^\circ \). 2. Recall that \( \tan x = \frac{\s

1. **State the problem:** Prove that $$\frac{\tan x + \sec x - 1}{\tan x - \sec x + 1} = \frac{1 + \sin x}{\cos x}.$$\n\n2. **Rewrite tangent and secant in terms of sine and cosine

1. **State the problem:** A boy observes a tree from two points on the ground. From the first point, the angle of elevation to the top of the tree is $25^\circ$. He then walks 30 m

1. **State the problem:** Prove that $$\frac{\sin^2 x - 1}{\tan x \sin x - \tan x} = \cos x + \cot x.$$\n\n2. **Rewrite the numerator:** Note that $$\sin^2 x - 1 = -(1 - \sin^2 x)

1. **State the problem:** Prove that $$\frac{1+2\cos x \sin x}{\cos x \sin x + \cos^2 x} = 1 + \tan x.$$\n\n2. **Rewrite the expression:** Start with the left-hand side (LHS): $$\f

1. **State the problem:** Simplify the expression $$\frac{1+2\cos x \sin x}{\cos x \sin x + \cos^2 x}$$. 2. **Rewrite the numerator:** The numerator is $$1 + 2\cos x \sin x$$.

1. **State the problem:** Use De Moivre's theorem to show that $$\frac{\sin 4\theta}{\sin \theta} = 8 \cos 3\theta - 4 \cos \theta.$$\n\n2. **Recall De Moivre's theorem:** For any

1. **State the problem:** Simplify the expression $$\frac{1-2\sin x \cos x}{1+2\sin x \cos x}$$. 2. **Recall the double-angle identity:** We know that $$\sin(2x) = 2\sin x \cos x$$

Problem: In triangle ABC, the angles satisfy $A=70^\circ$, $B=50^\circ$, and the side length $c=18$. Find the side $a$ to two decimal places. 1. The sum of the angles in a triangle

1. The problem is to solve the equation $2 \sin(x - 1) = 0$ for $x$. 2. First, divide both sides of the equation by 2 to isolate the sine function: