三角學：练习题

In triangle \( ABC \), side length \( AB = 7 \text{ cm} \), side length \( AC = 10 \text{ cm} \), and angle \( BAC = 150^\circ \). Calculate the area of the triangle.

A sector of a circle has a radius of \( 6 \text{ cm} \) and an arc length of \( 4 \text{ cm} \). Find the area of the sector.

Find all the solutions to the equation \( \sin 2\theta = \cos \theta \) in the interval \( 0 \le \theta < 2\pi \).

In triangle \( ABC \), the length of side \( AB = 7 \text{ cm} \), the length of side \( BC = 12 \text{ cm} \), and the size of angle \( ABC = 40^\circ \).
Calculate the area of triangle \( ABC \) in \( \text{cm}^2 \), giving your answer to 3 significant figures.

Solve the equation \( 6\cos^2 x + \sin x - 5 = 0 \) for the interval \( 0 \le x < 360^\circ \).
Give your answers to the nearest degree.

In triangle \( ABC \), the length of side \( AB = 8 \text{ cm} \), the length of side \( AC = 11 \text{ cm} \) and the angle \( BAC = 35^\circ \).
(a) Calculate the length of the side \( BC \), giving your answer to 3 significant figures.
(b) Find the area of triangle \( ABC \), giving your answer to 2 decimal places.

Solve, for \( 0 \le \theta < 360^\circ \), the equation
\( 4 \cos^2 \theta + 7 \sin \theta - 2 = 0 \)
Give your answers to one decimal place where appropriate.