📏 trigonometry

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

1. **State the problem:** Solve the equation $\sin x = \sin 282^\circ$ for $0^\circ \leq x \leq 270^\circ$. 2. **Recall the sine identity:** For angles $x$ and $a$, $\sin x = \sin

1. The problem asks us to find the values of $x$ in the interval $180^\circ \leq x \leq 360^\circ$ such that $\cos x = \cos 132^\circ$. 2. Recall the cosine function's property: $\

1. **State the problem:** Solve the equation $\tan x = \tan 38^\circ$ for $90^\circ \leq x \leq 360^\circ$. 2. **Recall the periodicity of tangent:** The tangent function has a per

1. Express $\tan 4x$ in terms of $\tan x$. Using the tangent multiple angle formula:

1. **State the problem:** Show that $ (\sin A + \cos A)^2 = 1 + \sin 2A $ and find the maximum value of $ 4(\sin A + \cos A)^2 $. 2. **Expand the left side:**

1. **State the problem:** Solve the equation $$15 \sin^2 x = 13 + \cos x$$ for $$0^\circ \leq x \leq 180^\circ$$. 2. **Rewrite the equation:** Use the Pythagorean identity $$\sin^2

1. **State the problem:** (i) Given the equation $$3 \sin^2 x - 8 \cos x - 7 = 0,$$ show that for real values of $$x,$$ $$\cos x = -\frac{2}{3}.$$

1. Problem 26(i): Prove the identity $$\frac{\cos \theta}{\tan \theta (1 - \sin \theta)} \equiv 1 + \frac{1}{\sin \theta}$$ for all valid $\theta$. 2. Start with the left-hand side

1. Problem Q.62: Simplify $$\sqrt{\sec^2\theta + \csc^2\theta} \times \sqrt{\tan^2\theta - \sin^2\theta}$$. Step 1: Use identities: $$\sec^2\theta = 1 + \tan^2\theta$$ and $$\csc^2

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Rewrite the tangent term:** Recall that $$\tan(\theta) = \frac{\sin

1. **State the problem:** Solve for $\theta$ in degrees the equation $$2\sin^2(\theta) - 3\sin(\theta) + 1 = 0$$ where $0^\circ \leq \theta < 360^\circ$. 2. **Substitute:** Let $x

1. The problem is to evaluate $\sin\left(\frac{\pi}{2}\right)$.\n\n2. Recall that $\sin(\theta)$ gives the y-coordinate of the point on the unit circle at angle $\theta$.\n\n3. The

1. The term "Sin invers" usually refers to the inverse sine function, also called arcsine. 2. The inverse sine function is denoted as $\sin^{-1}(x)$ or $\arcsin(x)$.