📏 trigonometry

1. The problem is to prove or simplify an expression involving $\sin^2(x) \times \tan^2(x)$ and show how it equals the left side of the equation. 2. Recall the identity for tangent

1. **State the problem:** Prove or disprove the identity $$\tan^2 \theta - \sin^2 \theta \stackrel{?}{=} \sin^2 \theta \tan^2 \theta$$. 2. **Recall definitions and formulas:**

1. **Énoncé du problème :** Calculer les valeurs exactes de $\sin\left(\frac{5\pi}{6}\right)$, $\cos\left(-\frac{13\pi}{6}\right)$, $\cos\left(\frac{5\pi}{3}\right)$, et $\sin\left

1. **Problem statement:** We have a tower with base point $P$ on horizontal ground. Points $A$ and $B$ lie on the same straight line with $P$, and the angles of elevation from $A$

1. **Problem statement:** Find the exact values of the following trigonometric ratios: (i) $\sin\left(\frac{5\pi}{6}\right)$

1. The problem is to find the value of $\cos 30^\circ$. 2. Recall the cosine of special angles: $\cos 30^\circ = \frac{\sqrt{3}}{2}$.

1. The problem is to find the value of $\cos \frac{1}{2}$. 2. The cosine function, $\cos x$, gives the horizontal coordinate of a point on the unit circle at an angle $x$ radians f

1. The problem asks which triangle shows the equation $\sin \Theta = \frac{15}{17}$. 2. Recall the definition of sine in a right triangle:

1. **State the problem:** We have a triangle formed by points A, B (the boat), and C. - Point C is 1.8 km east of B.

1. **Problem statement:** Find the exact value of $\cos L$ in simplest radical form for a right triangle $\triangle LKJ$ with right angle at $K$, sides $LK=2$, $KJ=6$, and hypotenu

1. The problem asks to find the exact value of $\cos LL$ in simplest radical form. 2. Since the problem does not specify what $LL$ represents, we need to clarify or assume it refer

1. **State the problem:** Prove that $$\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} = \csc \theta + \cot \theta$$. 2. **Recall identities:**

1. State the problem: Compute $\cos 30^{\circ}$. 2. Formula and rules: For a right triangle $\cos(\theta)=\dfrac{\text{adjacent}}{\text{hypotenuse}}$ and in a 30-60-90 triangle the

1. **State the problem:** We are given a sinusoidal function of the form $y = a \sin(bx)$ or $y = a \cos(bx)$ and a graph that oscillates between -4 and 4 on the y-axis with zeros

1. **State the problem:** We need to write an equation of the form $y = a \sin bx$ or $y = a \cos bx$ that describes the given graph. 2. **Analyze the graph:** The graph is a cosin

1. The problem asks to calculate $$\frac{9}{\sin^4 \alpha + \cos^4 \alpha}$$ given that $$\cot \alpha = \sqrt{3}$$. 2. Recall that $$\cot \alpha = \frac{\cos \alpha}{\sin \alpha}$$

1. The problem asks to find the value of $$\frac{\sin \alpha + \cos \alpha}{\sin \alpha - \cos \alpha}$$ given that $$\tan \alpha = \frac{4}{5}$$. 2. Recall the identity $$\tan \al

1. The problem asks to solve the equation $$\frac{2 \sin x - \cos x}{2 \cos x + \sin x} = 3$$ for $x$. 2. We start by multiplying both sides by the denominator to clear the fractio

1. Énoncé du problème : Trouver toutes les valeurs de $x$ dans l'intervalle $[0, \pi]$ telles que $\sec 3x = 3$. 2. Rappel de la définition : $\sec \theta = \frac{1}{\cos \theta}$.

1. **State the problem:** Solve the equation $$\csc\theta - 1 = \frac{\cot^2\theta}{\csc\theta + 1}$$ for $\theta$. 2. **Recall identities:**

1. **State the problem:** Solve the equation $$6 \cos^2 \theta - 4 = \sin \theta$$ for $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Use the Pythagorean identity:** Recall that $$\c