📏 trigonometry

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:** $$\sin^2 x + \cos^2 x = 1$$.

1. **State the problem:** Simplify the expression $$\frac{1-\sin^4 x}{\cos^4 x}$$. 2. **Recall the formula and identities:** Use the difference of squares formula: $$a^2 - b^2 = (a

1. The problem is to find the value of $\theta$ given some context or equation involving $\theta$. 2. To solve for $\theta$, we need an equation or expression where $\theta$ is the

1. The problem is to find the value of $\cos\theta$ for a given angle $\theta$. 2. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent

1. Problem: Find the value of $\theta$ if $\tan \theta = 1$. The tangent function equals 1 at angles where the opposite and adjacent sides are equal. The principal angle is $\theta

1. **Evaluate $\tan(\cos(-\frac{11\pi}{2}))$** - First, find $\cos(-\frac{11\pi}{2})$.

1. The problem is to evaluate $\cos(\cos 270^\circ)$.\n\n2. First, recall that $\cos 270^\circ$ means the cosine of 270 degrees.\n\n3. Using the unit circle, $\cos 270^\circ = 0$.\

1. **State the problem:** Find the exact value of $\csc\left(-\frac{5\pi}{6}\right)$.\n\n2. **Recall the definition:** $\csc(\theta) = \frac{1}{\sin(\theta)}$. So we need to find $

1. **State the problem:** Evaluate $\cot 300^\circ$. 2. **Recall the definition:** $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

1. The problem is to evaluate $\tan 240^\circ$. 2. Recall that the tangent function is periodic with period $180^\circ$, and $\tan(\theta) = \frac{\sin \theta}{\cos \theta}$.

1. The problem asks: In which quadrants is the sine function negative? 2. Recall that the sine function, $\sin(\theta)$, represents the y-coordinate of a point on the unit circle a

1. **State the problem:** Simplify and verify the identity $$\frac{\tan \alpha}{\sec \alpha - 1} = \frac{\sec \alpha + 1}{\tan \alpha}$$. 2. **Recall basic trigonometric identities

1. **State the problem:** Prove the trigonometric identity $$\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$$. 2. **Recall the sum-to-produ

1. **State the problem:** Simplify the expression $$\frac{(\sin x)^3 - (\cos x)^3}{\sin x + \cos x} = \frac{(\csc x)^2 - \cot x + 2(\cos x)^2}{1 - (\cot x)^2}$$. 2. **Recall formul