Question

\(\sin \left(-420^{\circ}\right) \cos \left(390^{\circ}\right)+\cos \left(-660^{\circ}\right) \sin \left(330^{\circ}\right)\) is (a) 1 (b) \(-1\) (c) 2 (d) \(-2\)

Short answer

Answer: (b) -1

Step-by-step solution

Find equivalent angles

To find the equivalent angles for each trigonometric function, we will use the fact that both sin and cos functions are periodic with a period of \(360^{\circ}\). We can add or subtract multiples of \(360^{\circ}\) to any angle without changing the value of the trigonometric function. Equivalent angles: \(\sin(-420^{\circ}) = \sin(-420^{\circ} + 360^{\circ}) = \sin(-60^{\circ})\) \(\cos(390^{\circ}) = \cos(390^{\circ} - 360^{\circ}) = \cos(30^{\circ})\) \(\cos(-660^{\circ}) = \cos(-660^{\circ} + 720^{\circ}) = \cos(60^{\circ})\) \(\sin(330^{\circ}) = \sin(330^{\circ} - 360^{\circ}) = \sin(-30^{\circ})\)

Determine sin and cos values

Now that we have found the equivalent angles, let's find the sin and cos values for these angles. Using the values for 30, 60, and their respective multiples: \(\sin(-60^{\circ}) = -\sin(60^{\circ}) = -\frac{\sqrt{3}}{2}\) (Since sine is an odd function) \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\) \(\cos(60^{\circ}) = \frac{1}{2}\) \(\sin(-30^{\circ}) = -\sin(30^{\circ}) = -\frac{1}{2}\) (Since sine is an odd function)

Perform the addition

Now that we have the sin and cos values, we can perform the addition: \(= (-\frac{\sqrt{3}}{2})\frac{\sqrt{3}}{2} + (\frac{1}{2})(-\frac{1}{2})\) \(= (-\frac{3}{4}) + (-\frac{1}{4})\) \(= -\frac{3 + 1}{4}\) \(= -\frac{4}{4}\) \(= -1\)

Match the result with the options

Now, let's match our result with the given options: Our result is -1, which corresponds to option (b). Therefore, the answer to the exercise is (b) -1.

Key concepts

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics, especially in geometry and trigonometry. They relate the angles of a triangle to the ratios of its sides. These functions are periodic, repeating their values in a specific cycle. The sine function, \(\sin(\theta)\), measures the vertical component of the angle \(\theta\). Cosine, \(\cos(\theta)\), measures the horizontal component. These two are especially important as they can express other trigonometric functions like tangent, which is the quotient of sine and cosine.

• Sine (\(\sin\): Measures the y-component in a unit circle.
• Cosine (\(\cos\): Measures the x-component in a unit circle.
• Periodicity: cyclic behavior with a period of \(360^{\circ}\) for sine and cosine.

Understanding the properties and interrelationships of these functions helps solve a wide range of problems.

Angle Conversion

Angles in trigonometry can be represented in degrees or radians. Degrees are more common in geometry and everyday usage, while radians are often used in calculus because they provide a direct measure of the angle in terms of pi. Converting angles involves understanding how many degrees are in a circle (360°) and how that translates to radians (2π).To convert an angle from degrees to radians, multiply by \(\frac{\pi}{180}\)\.

• Example: Convert 180° to radians: \(180^{\circ} \times \frac{\pi}{180} = \pi \text{ radians}\)
• Example: Convert -420°: first simplify by adding 360°, resulting in -60°.

Understanding angle conversion is crucial, especially in fields such as physics and engineering, where precise measurements are needed.

Periodicity

Periodicity is an essential concept in trigonometry. It refers to the repeating nature of trigonometric functions over specific intervals. For sine and cosine, this interval is \(360^{\circ}\) or \(2\pi\) radians. This means that adding or subtracting \(360^{\circ}\) or \(2\pi\) from an angle doesn't change the functional value.Periodic functions allow us to simplify complex angles:

• Example 1: \(-420^{\circ} + 360^{\circ} = -60^{\circ}\)
• Example 2: Modifying angles like 390° to their equivalent 30° through subtraction.

Recognizing and applying periodicity reduces angles to a principal range, typically \(0^{\circ} - 360^{\circ}\) or \(0 - 2\pi\) radians, making calculations more straightforward.

Sine and Cosine Values

Finding the exact sine and cosine values of standard angles, such as 30°, 45°, 60°, and their negative counterparts, is a common task in trigonometry. Sine values can be negative, positive, or zero depending on the angle's position (quadrant) on the unit circle.Here are some standard sine and cosine values:

• \(\sin(30^{\circ}) = \frac{1}{2}\): The sine value for 30°.
• \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\): The cosine value for 30°.
• Odd function property of sine: \(\sin(-\theta) = -\sin(\theta)\), meaning \(\sin(-30^{\circ}) = -\frac{1}{2}\).
• Even function property of cosine: \(\cos(-\theta) = \cos(\theta)\), so \(\cos(-60^{\circ}) = \frac{1}{2}\).

Memorizing these standard values and understanding their properties are key to solving trigonometric problems efficiently.