📏 trigonometry

1. The problem is to evaluate the expression $$\cos^{-1}\left(\cos\left(\frac{7\pi}{8}\right)\right).$$ 2. Recall that the function $$\cos^{-1}(x)$$, also called arccosine, returns

1. **State the problem:** Simplify the expression $$m = \frac{\sqrt{6} \sec 45^\circ \sec 30^\circ + 5 (\sin 37^\circ + \sin 53^\circ)}{\tan 45^\circ + 3 \sec 53^\circ}$$. 2. **Rec

1. **State the problem:** Simplify the expression $$R = \frac{(\sin \alpha + \cos \alpha)^2 + (\sin \alpha - \cos \alpha)^2}{(\sec \alpha + \cos \alpha)^2 - (\cos \alpha - \sec \al

1. **State the problem:** We need to find the tangent of angle $S$ in a right triangle with sides $TS=10$ (opposite to $S$), $TU=24$ (adjacent to $S$), and hypotenuse $SU=26$. 2. *

1. **State the problem:** Prove or simplify the trigonometric identity $$\frac{1 + \csc x}{\sec x} = \cos x + \cot x$$. 2. **Recall definitions and formulas:**

1. **State the problem:** Prove the identity $$(1 + \tan^2 u)(1 - \sin^2 u) = 1.$$\n\n2. **Recall important formulas:**\n- Pythagorean identity: $$1 + \tan^2 u = \sec^2 u.$$\n- Pyt

1. **State the problem:** We need to find the horizontal distance $d$ between the observation tower and the fire.

1. **Problem:** Find the value of side AB in a triangle where angle B = 50°, angle C = 62°, and side AC = 12. 2. **Step 1: Find angle A.**

1. The problem is to find the value of $\sin(\pi)$. 2. Recall the unit circle definition: $\sin(\theta)$ is the y-coordinate of the point on the unit circle at angle $\theta$ radia

1. We need to prove the trigonometric identity: $$\frac{1-\sin^4\alpha}{\cos^2\alpha} - 2\sin^2\alpha = \cos^2\alpha$$. 2. Recall the Pythagorean identity: $$\sin^2\alpha + \cos^2\

1. **State the problem:** Given $\cos A = \frac{3}{5}$ with $A$ acute, and $\sin B = \frac{5}{13}$ with $B$ obtuse, find $\cos(A-B) - \sin(A-B)$ without calculators. 2. **Recall fo

1. The problem is to find the values of $x$ for which $\cos x < 0$. 2. Recall that the cosine function $\cos x$ is negative in the intervals where the angle $x$ lies in the second

1. **State the problem:** Evaluate the expression $$\cos\left(-\frac{\pi}{4}\right) + \sin\left(-\frac{7\pi}{6}\right)$$. 2. **Recall the relevant formulas and properties:**

1. The problem asks us to evaluate the expression $\sin(135^\circ) - \cos(45^\circ)$.\n\n2. Recall the values of sine and cosine for special angles: \n- $\sin(135^\circ) = \sin(180

1. **Convert degrees to radians:** Use the formula $\text{radians} = \text{degrees} \times \frac{\pi}{180}$.

1.1 Show that $\sin \alpha = \cos(90^\circ - \alpha)$. 1. Recall the complementary angle identity: $\sin \theta = \cos(90^\circ - \theta)$.

1. **State the problem:** Simplify the expression $$\frac{\sin x \cdot \cos x}{\cos n x}$$ where $x$ and $n$ are variables. 2. **Recall relevant formulas:** There is no direct simp

1. **Problem Statement:** Simplify the trigonometric expressions without using a calculator. 2. **Formula and Rules:** Use angle sum/difference identities and reference angles in t

1. **State the problem:** We have two flying discs detected at the same angle of elevation $39.48^\circ$ but at different distances from the receiver: 826 m and 1296 m. We need to

1. **State the problem:** We are given two sinusoidal functions $y=f(x)$ and $y=g(x)$ graphed on a Cartesian coordinate system with $x$ in degrees and $y$ values ranging approximat

1. **State the problem:** We have a kite flying such that from point A on the ground, the string makes a 75° angle of elevation, and from point B, which is 240 ft horizontally from