PCCP Practice Midterm Exam

Front

What is the Law of Sines?

Back

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. It is expressed as: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).

Front

What is the Law of Cosines?

Back

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful for finding a side or angle when you have two sides and the included angle or all three sides. It is expressed as: \( c^2 = a^2 + b^2 - 2ab \cos(C) \).

Front

How do you convert degrees to radians?

Back

To convert degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \). For example, to convert 180 degrees to radians: \( 180 \times \frac{\pi}{180} = \pi \) radians.

Front

What is the range of the cosine function?

Back

The range of the cosine function is \([-1, 1]\). This means that the value of \( \cos(\theta) \) will always be between -1 and 1 for any angle \( \theta \).

Front

What is the general solution for \( \cos \theta = k \)?

Back

The general solution for \( \cos \theta = k \) is given by: \( \theta = \pm \cos^{-1}(k) + 2n\pi \), where \( n \) is any integer.

Front

What is the unit circle?

Back

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It is used to define trigonometric functions and their values for different angles.

Front

How do you find the sine of an angle using the unit circle?

Back

To find the sine of an angle using the unit circle, locate the point on the circle corresponding to that angle. The y-coordinate of that point gives the value of \( \sin(\theta) \).

Front

What is the relationship between radians and the unit circle?

Back

In the unit circle, the angle in radians corresponds to the length of the arc on the circle. For example, an angle of 1 radian subtends an arc length of 1 unit on the circle.

Front

What is the sine function's range?

Back

The range of the sine function is \([-1, 1]\). This means that the value of \( \sin(\theta) \) will always be between -1 and 1 for any angle \( \theta \).

Front

How do you solve \( 4\cos\theta - 2\sqrt{3} = 0 \)?

Back

To solve \( 4\cos\theta - 2\sqrt{3} = 0 \), first isolate \( \cos\theta \): \( \cos\theta = \frac{\sqrt{3}}{2} \). The solutions in the interval \( [0, 2\pi) \) are \( \theta = \frac{\pi}{6}, \frac{11\pi}{6} \).