📏 trigonometry

1. The problem is to prove the identity: $$\cos 2A - \cot 2A = \tan 2A$$. 2. Recall the definitions and identities:

1. **State the problem:** Prove or simplify the expression $$\cos 2A - \cot 2A = \tan A$$. 2. **Recall formulas:**

1. **Problem statement:** (i) Show that $8 \sin^2 x \cos^2 x$ can be written as $1 - \cos 4x$.

1. The problem is to find the value of $\cos\left(\frac{2\pi}{3}\right)$.\n\n2. Recall that the cosine function for an angle $\theta$ in radians gives the x-coordinate of the point

1. Énoncé du problème : Résoudre l'équation trigonométrique $$\sin(x) = \frac{1}{2}$$ sur l'intervalle $$[0, 2\pi]$$. 2. Formule utilisée : Pour résoudre $$\sin(x) = a$$, on utilis

1. **State the problem:** Solve the trigonometric equation $$\cos 2A - \cot 2A = \tan A$$ for angle $A$. 2. **Recall formulas and identities:**

1. **State the problem:** Prove or verify the identity $$\frac{\sin A + \sin 2A}{1 + \cos A + \cos 2A} = \tan A.$$\n\n2. **Recall formulas:** Use the double-angle formulas: $$\sin

1. The problem is to verify the identity: $\sin^4 b + \cos^4 b = 1 - \frac{1}{2} \sin^2 2b$. 2. Recall the Pythagorean identity: $\sin^2 b + \cos^2 b = 1$.

1. **State the problem:** Romeo and Paris are observing Juliet's balcony from two different positions. Romeo faces north and sees the balcony at an angle of elevation of 20 degrees

1. **State the problem:** Solve the trigonometric equation $$\cos 2x - \cos x = 0$$ for $x$. 2. **Recall the double-angle formula:** $$\cos 2x = 2\cos^2 x - 1$$.

1. Let's start by understanding what trigonometry is. Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially

1. **Problem statement:** We have triangle ABC on horizontal ground with points A, B, and C. - Bearing of C from A is 145°.

1. **Problem statement:** Find the exact values of:

1. **State the problem:** Given that $\sin A = \frac{5}{3}$ and $A$ is in the second quadrant, find $\cos 2A$ and $\sin 2A$. 2. **Note:** The value $\sin A = \frac{5}{3}$ is not po

1. **Problem statement:** Given $\cos A = \frac{\sqrt{3}}{2}$, find the value of $\cos 2A$. 2. **Formula used:** The double-angle formula for cosine is:

1. **State the problem:** We need to find the height $h$ of the tower given a right triangle where the angle between the hypotenuse and the horizontal base is $45^\circ$, the horiz

1. **State the problem:** Solve the equation $$\cos(x + 30^\circ) - \cos(x + 48^\circ) = 0.2$$ for $$30^\circ \leq x \leq 360^\circ$$. 2. **Use the cosine difference identity:**

1. **Problem statement:** Leah and Malia stand on opposite sides of a church spire. Leah is 120 m from the spire with an angle of elevation of 18° to the top. Malia's angle of elev

1. **Problem statement:** A jetliner is flying at 35,000 feet above the ocean. The pilot measures the angle of depression to the coast of an island as 5°. We need to find the horiz

1. **Problem statement:** Shawn measures the angle of elevation to the top of a 450 feet high tower as 32°. We need to find the horizontal distance from Shawn to the base of the to

1. **Problem statement:** We have a ladder leaning against a wall forming a right triangle. The foot of the ladder is 2 m from the wall, and the angle between the ladder and the gr