📏 trigonometry

1. The problem is to solve the trigonometric equation $$\sin(x) = \frac{1}{2}$$ for all solutions. 2. From the unit circle, we know that $$\sin(x) = \frac{1}{2}$$ at two principal

1. **Problem statement:** In triangle $\triangle PQR$, given $\angle Q = 42^\circ$, side $PR = 12$ cm, and side $PQ = 10.2$ cm, find: (i) $\angle R$

1. The problem is to convert the angle $\frac{2 \pi}{3}$ radians into degrees. 2. Recall the conversion formula between radians and degrees:

1. The problem is to convert an angle of 150° to radians. 2. Recall the conversion formula between degrees and radians: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

1. You asked to work with angles in radians. 2. Radians are a way to measure angles based on the radius of a circle.

1. The problem is to find the value of $\cos\left(\frac{\sqrt{11}}{6}\right)$.\n\n2. Here, $\frac{\sqrt{11}}{6}$ is the angle in radians. We need to evaluate the cosine of this ang

1. **State the problem:** Given $\cos \theta = \sqrt{\frac{11}{6}}$ and $\theta$ is in Quadrant I, find the exact value of $\sin \theta$ in simplest form. 2. **Recall the Pythagore

1. We are given that $\cos \theta = -\frac{1}{6}$ and that $\theta$ is in Quadrant III. 2. In Quadrant III, both sine and cosine are negative.

1. **State the problem:** Given $\cos \theta = -\frac{2}{3}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin \frac{\theta}{2}$, $\cos \frac{\theta}{2}$, and $\tan \frac{\theta}{2}$

1. **Evaluate the expression:** $$8 \sin 45^\circ - \sin 60^\circ \cdot \cos 30^\circ - \frac{1}{4} \tan 45^\circ$$ 2. **Recall exact trigonometric values:**

1. **Problem 3.4:** Determine the angle $\theta$ formed by the line from the origin to the point $(3, -4)$ with the positive x-axis, correct to 2 decimal places. 2. The point $(3,

1. The problem involves understanding the inverse tangent function, often written as $\tan^{-1}(x)$ or $\arctan(x)$, which gives the angle whose tangent is $x$. 2. To find $\arctan

1. **Problem:** Find $\sin^{-1}(\frac{1}{2})$. Step 1: Recall that $\sin^{-1}(x)$ is the angle whose sine is $x$.

1. The problem asks to calculate $\tan^{-1}\left(\frac{r}{15}\right)$ for each given value of $r$.\n\n2. Recall that $\tan^{-1}(x)$, also called arctangent, is the inverse function

1. The problem is to simplify the expression $\tan x \cdot \sec x$. 2. Recall the definitions: $\tan x = \frac{\sin x}{\cos x}$ and $\sec x = \frac{1}{\cos x}$.

1. The problem involves finding the distance between points X and Y given their bearings and distances from point O. 2. Point X is located 40 m from O at a bearing of 047° clockwis

1. **State the problem:** Simplify the expression $$\frac{1-\sin^2 x}{\cos x}$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^2 x = 1$$.

1. **State the problem:** We have triangle ABC with sides AC = 8 cm, BC = 15 cm, and angle ACB = 70°.

1. The problem is to find angle $C$ in a triangle using the sine rule, then calculate the remaining angle using $180 - 40 + C$. 2. The sine rule states: $$\frac{a}{\sin A} = \frac{

1. **Understanding the basic trigonometric functions:** The primary functions are sine ($\sin x$), cosine ($\cos x$), and tangent ($\tan x$). Each has a characteristic wave shape a

1. **Stating the problem:** We have a right triangle with a vertical leg of length 21 km and an angle of 142° given. We want to find the length of the hypotenuse or other sides if