📏 trigonometry

1. **Problem Statement:** We need to find the correct ratio for $\cos C$ in a right triangle with vertices A, B, and C, where the right angle is at B. The sides opposite vertices A

1. **Problem Statement:** Determine the values of $a$, $b$, and $c$ in the function $f(x) = a \cos(bx) + c$ for $0^\circ \leq x \leq 360^\circ$, given the graph oscillates between

1. **State the problem:** Given a triangle with side $b=8.5$ inches, side $c=18.4$ inches, and angle $A=53.213^\circ$, solve for the remaining side $a$ and angles $B$ and $C$. 2. *

1. **State the problem:** We are given the ratio \( \frac{\sin 50}{u} = \frac{\sin 55}{x} \) and the value \( u = 11.8 \). We want to find \( x \).

1. **State the problem:** Solve the equation $$\frac{8}{f} = \cos 46^\circ$$ for $f$ and give the answer to 2 decimal places. 2. **Recall the formula and rules:** The equation rela

1. The problem is to calculate $\cos(75^\circ) \times 10$ and round the result to 2 decimal places. 2. Recall the cosine of an angle in degrees can be found using a calculator or t

1. Let's start by evaluating the trigonometric expressions given. 2. (a) Evaluate $\cot 42^\circ \times \sec 105^\circ$ and check if it equals $\csc 15^\circ$.

1. **State the problem:** Given a point $P = (8, -6)$ on the terminal side of angle $\theta$, find $\sin \theta$. 2. **Recall the formula:** For a point $(x, y)$ on the terminal si

1. The problem asks for the reciprocal of $\sin\theta$. The reciprocal of a trigonometric function $f(\theta)$ is $\frac{1}{f(\theta)}$. For $\sin\theta$, the reciprocal is $\csc\t

1. **Problem Statement:** Identify the equation that represents the given graph. 2. **Graph Description:** The graph oscillates between $-1$ and $1$, crosses the y-axis at $y=1$, a

1. **State the problem:** Calculate $\tan\left(\frac{92\pi}{3}\right)$.\n\n2. **Recall the periodicity of tangent:** The tangent function has a period of $\pi$, meaning $\tan(x) =

1. Probleem 3.1: Bereken die waarde van $\sin^2 27^\circ + 2 \cos 54^\circ$. Formule: Gebruik die definisies van sinus en kosinus en die feit dat $\sin^2 x = (\sin x)^2$.

1. **Problem 1: Calculate the hypotenuse of a right-angled triangle** Given: angle $26^\circ$, opposite side length $18.6$ cm.

1. **State the problem:** Simplify and verify the identity $$\frac{\tan(a)-\cot(a)}{\sin(a)\cos(a)} = \sec(a)(-\csc^2(a))$$. 2. **Recall definitions and formulas:**

1. 問題陳述： (1) 已知 $90^\circ < \alpha, \beta, \gamma < 180^\circ$，且 $\tan \alpha = -2$, $\tan \beta = -\frac{1}{3}$, $\tan \gamma = -\frac{1}{7}$，求 $\alpha + \beta + \gamma$。

1. Problem statement: Prove or simplify the identity $\frac{1-\cos(2\theta)}{\sin(2\theta)} + \frac{\sin(2\theta)}{1-\cos(2\theta)} = \frac{2}{\tan\theta}$. 2. Formulas and rules u

1. Problem 3.1: Given $\hat{A} = 20^\circ$ and $\hat{B} = 35^\circ$, calculate $3\sqrt{\sec A \cdot \csc^3 B}$. Formula: $\sec \theta = \frac{1}{\cos \theta}$ and $\csc \theta = \f

1. **Problem 09:** Given $\tan A = 2$ and $\tan B = 3$, find $\frac{\cos(A+B)}{\sin(A-B)}$. 2. Recall the formulas:

1. **State the problem:** Given $\sin \alpha = \frac{3}{5}$ with $\alpha$ in quadrant II, and $\tan \beta = \frac{5}{12}$ with $\beta$ in quadrant III, find $\cos(\alpha + \beta)$.

1. **Problem Statement:** We have a right triangle formed by a planet, its moon, and its sun. The distance between the planet and its moon is 141,000 km, and the angle at the moon

1. **State the problem:** Find the exact value of $\tan 165^\circ$ using a sum or difference formula. 2. **Recall the formula:** The tangent of a sum or difference of angles is giv