📏 trigonometry

1. **Problem statement:** We have triangle ABC with sides |AB| = 5, |AC| = 3, and angles |\angle ABC| = W, |\angle ACB| = 2W, where $0^\circ < W < 45^\circ$. We want to find $\cos

1. **Problem Statement:** Find the angle of \(\angle HAM\) and the length of segment \(MT\) in triangle \(HAM\) where \(AM = 26\) cm, \(HM = 46\) cm, and \(\angle M\) is split into

1. The problem is to verify if the solutions to a trigonometric equation are $\frac{7\pi}{6}$, $\frac{11\pi}{6}$, and $\frac{\pi}{2}$. 2. Typically, such solutions come from equati

1. **State the problem:** Solve the trigonometric equation $\sin^2 x - \cos^2 x - \sin x = 0$. 2. **Use the Pythagorean identity:** Recall that $\sin^2 x + \cos^2 x = 1$, so $\cos^

1. **State the problem:** We need to find the exact value of $\tan \theta$ where $\theta$ is an angle in standard position and its terminal side passes through the point $(-1, -1)$

1. **State the problem:** Given that $\sin A = -\frac{20}{29}$ and angle $A$ is in Quadrant III, find the exact value of $\tan A$ in simplest radical form with a rational denominat

1. **State the problem:** We have a right triangle with hypotenuse $R=5$ and need to find the value of $r$, the trigonometric ratios $\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$,

1. **Problem statement:** Find the length of the missing side in each right triangle given the angles and the right angle.

1. **State the problem:** We are given the function $$d(t) = 3 \sin\left(\frac{2\pi}{365}(t - 79)\right) + 12$$ which models the number of hours of daylight on day $$t$$ of the yea

1. **State the problem:** Given triangle ABC with angles $A=72^\circ$, $B=43^\circ$, and side $a=23$ opposite angle $A$, find angle $C$ and sides $b$ and $c$. 2. **Find angle $C$:*

1. **State the problem:** We need to find the exact values of $\cot \theta$, $\sin \theta$, and $\csc \theta$ for the angle $\theta$ in a right triangle where the legs adjacent to

1. **State the problem:** Solve the equation $\sin 5x - 1 = 0$ for $x$. 2. **Rewrite the equation:** Add 1 to both sides to isolate the sine term:

1. Staðfesta vandamálið: Finna gildi fyrir $a \cos v$ þegar $a$ og $v$ eru gefin. 2. Formúla: $a \cos v$ þýðir margfeldi $a$ og kósínusar hornsins $v$.

1. Við erum beðin um að finna $v$ gefið að $\sin v = \frac{4}{10}$ og að $v$ sé í fyrsta fjórðungi. 2. Í fyrsta fjórðungi eru bæði $\sin v$ og $\cos v$ jákvæð.

1. **State the problem:** Given an acute angle $\theta$ in a right triangle with $\sin \theta = \frac{1}{3}$, find $\cos \theta$. 2. **Recall the Pythagorean identity:**

1. **State the problem:** Solve the inequality $$\sin\left(x+\frac{\pi}{3}\right) - \cos\left(x+\frac{\pi}{6}\right) > -\frac{1}{2}.$$\n\n2. **Use trigonometric identities:** Recal

1. **State the problem:** Solve the equation $$\cos\left(x-\frac{\pi}{6}\right) + \cos\left(x+\frac{\pi}{6}\right) = \frac{3}{2}$$ for $x$. 2. **Use the cosine sum formula:** Recal

1. **State the problem:** We have a right triangle with sides 4 (vertical), 4\sqrt{3} (horizontal), and 8 (hypotenuse), and angles 30° and 60°. We want to find \(\sin 60^\circ\) an

1. **Problem statement:** Given the equation $$13 \cos \alpha - 5 = 0$$ and the condition $$\tan \alpha < 0$$, find the value of $$13 \sin \alpha + 25 \tan^2 \alpha$$ without using

1. **Problem Statement:** Determine the value of $k$ using the given diagram (top-right coordinate graph with angle $\theta$ and line segment labeled 41).

1. **State the problem:** Simplify the expression $$\frac{\cos^2 a - \sin^2 a}{\cos^2 a (2 - \cos^2 a)}$$. 2. **Recall the formula:** Use the Pythagorean identity and double-angle