📏 trigonometry

1. **State the problem:** Solve the equation $$\sin 2x = 2 \sin x$$ for all solutions in the interval $$[0,2\pi)$$. 2. **Recall the double-angle formula:** $$\sin 2x = 2 \sin x \co

1. **State the problem:** We need to find the angle $x$ in a right triangle where the hypotenuse is 5 and the adjacent side to angle $x$ is 3. 2. **Recall the trigonometric formula

1. **Problem:** Find the missing side $x$ in the first right-angled triangle where the hypotenuse is 5 cm and the angle adjacent to the base is 27°. 2. **Formula:** Use the sine fu

1. **State the problem:** We are given the distance from a point P on the ground to the top of a tree as 6 m, and the angle of elevation from point P to the top of the tree is 59°.

1. **Problem:** Given a point on the terminal side of an angle $\theta$, find the six trigonometric functions: $\sin\theta$, $\cos\theta$, $\tan\theta$, $\csc\theta$, $\sec\theta$,

1. **State the problem:** Solve the trigonometric equation $$\tan\left(\frac{\pi}{5}x\right) - 1 = 0$$ for $$0 \leq x \leq 8$$. 2. **Rewrite the equation:** We want to find $$x$$ s

1. **State the problem:** Solve the equation $$3\sin^2\theta + 9\sin\theta = 0$$ for $$0 \leq \theta < 2\pi$$. 2. **Rewrite the equation:** Let $$x = \sin\theta$$. The equation bec

1. The problem is to simplify the expression $\tan\left(\frac{\pi}{4} + x\right)$.\n\n2. We use the tangent addition formula: $$\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan

1. We need to prove the identity $$\frac{1+\sin(2x)}{\cos(2x)} = \tan\left(\frac{\pi}{4} x\right)$$. 2. Recall the double-angle formulas:

1. **State the problem:** Simplify the expression $$\frac{1-\cos x}{\sin x} \times \frac{1}{\cos x}$$. 2. **Recall the formula and rules:** We will use trigonometric identities and

1. **Problem Statement:** We have a triangle with vertices T, B, and F. Given:

1. **State the problem:** We need to find the distance between yachts B and C (denoted as $a = BC$) using the given triangle with sides $b = 500$ m, $c = 700$ m, and angle $A = 15^

1. **Problem statement:** Given $f(x) = 3 \cos x - 4 \sin x$ and that $f(x) = R \cos(x + \alpha)$ with $R > 0$ and $0 \leq \alpha \leq 90^\circ$, find $R$ and $\alpha$. 2. **Formul

1. **Problem:** Find the exact value of $\sin(225^\circ)$. 2. **Formula and rules:** The sine of an angle in the unit circle can be found using reference angles and the signs in ea

1. **State the problem:** We need to find the values of $\cos x$ and $\cos z$ in a triangle with sides 7 m, 5 m, and 3 m, where angle $x$ is opposite the side of length 3 m, and an

1. **Stating the problem:** Given an angle $\alpha = 25^\circ$ and a distance $s = 45$ m, we want to find the height or vertical component related to these values. 2. **Formula use

1. The problem is to convert the given angle from degrees to radians, leaving the answer in terms of $\pi$. 2. The formula to convert degrees to radians is:

1. **State the problem:** We need to find the height $h$ of the tower. The triangle is right-angled with a horizontal side of 125 m, a vertical side of 1.8 m (man's height), and an

1. **State the problem:** A surfer is riding a 7-foot wave, and the angle of depression from the surfer to the shoreline is 10°. We need to find the distance from the surfer to the

1. **State the problem:** We are given three right triangle problems involving tangent ratios: - Find $x$ in $\tan 55^\circ = \frac{x}{10}$.

1. **State the problem:** Find the exact value of $\sec^{-1}(1)$ in radians. 2. **Recall the definition:** The inverse secant function $\sec^{-1}(x)$ gives the angle $\theta$ such