三角學：练习题

In triangle \(ABC\), the lengths of sides \(AB\) and \(AC\) are \(12 \text{ cm}\) and \(7 \text{ cm}\) respectively. Given that angle \(BAC = 150^\circ\), find the area of the triangle.

The diagram shows a sector of a circle. Use the values given in the diagram to find the radius, \(r\), of the circle.

Given that \(\sin^2 \theta = \frac{1}{4}\), find the value of \(\cos^2 \theta\).

A sector of a circle has a radius of \(10 \text{ cm}\) and a central angle of \(0.8\) radians. Calculate the length of the arc of this sector.

Find the values of \(x\) in the interval \(0 \le x < 360^\circ\) that satisfy the equation \(2 \sin x + 1 = 0\).

Given that \( \cos \theta = -\frac{12}{13} \) and \( 180^\circ < \theta < 270^\circ \), find the exact value of \( \sin \theta \).

Solve the equation \( 2\sin(2x) = 1 \) for the interval \( 0 \le x \le \pi \), giving your answers in terms of \( \pi \).

Find all values of \( \theta \) in the interval \( 0 \le \theta < 360^\circ \) that satisfy the equation \( 6\cos^2 \theta + \sin \theta - 5 = 0 \).

The diagram shows a sector of a circle with center \( O \) and radius \( r = 6 \text{ cm} \). The arc length of the sector is \( 9 \text{ cm} \) and the central angle is \( \theta \) radians.

(a) State the value of \( \theta \).
(b) Calculate the area of the sector.

In triangle \(ABC\), the side length \(AB = x\) cm, \(BC = (x + 1)\) cm, and \(AC = 7\) cm. Given that the angle \(\angle ABC = 120^\circ\):
(a) Form an equation in terms of \(x\) and show that it simplifies to \(3x^2 + 3x - 48 = 0\).
(b) Solve the equation to find the value of \(x\).
(c) Calculate the area of triangle \(ABC\), giving your answer in the form \(k\sqrt{3}\) where \(k\) is a constant.