Problem 40
Question
In Exercises 37 - 42, find the exact value of the expression. \( \cos 120^\circ \cos 30^\circ + \sin 120^\circ \sin 30^\circ \)
Step-by-Step Solution
Verified Answer
The exact value of the given trigonometric expression is 0.
1Step 1: Identify the identity
The given expression is in the form of a trigonometric identity. It is the formula for the cosine of difference of two angles; \(\cos(A - B) = \cos A \cos B + \sin A \sin B\). Here, A = 120 degrees and B = 30 degrees.
2Step 2: Replace in the identity
Now replace the given expression using the value A - B. This changes the expression to \(\cos(120^\circ - 30^\circ)\). Calculate the difference inside the cosine; 120 - 30 equals 90 degrees. Thus, the expression simplifies to \(\cos 90^\circ\).
3Step 3: Find the value of the cosine
The exact value of \(\cos 90^\circ\) is 0 in the unit circle. Thus, the entire expression evaluates to 0.
Key Concepts
Cosine of Difference of Two AnglesTrigonometric FunctionsUnit Circle
Cosine of Difference of Two Angles
Understanding the cosine of the difference of two angles is crucial for solving various trigonometric problems. In essence, this identity reveals how the cosine of the subtraction between two angles relates to the individual cosines and sines of these angles. The identity is expressed as
\[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \]
When confronted with an expression like \( \cos 120^\circ \cos 30^\circ + \sin 120^\circ \sin 30^\circ \), you can recognize it as conforming to this identity, with \( A = 120^\circ \) and \( B = 30^\circ \).
By applying the identity, you transform the original compound expression into a simple function: \( \cos(120^\circ - 30^\circ) \) or \( \cos 90^\circ \). This simplification makes it much easier to evaluate the expression, as you only need to find the cosine of a single angle.
There are a few common mistakes to avoid when using this identity. Firstly, ensure that you use the correct signs – note that after the cosine, it's a plus sign connecting the two products. Secondly, make sure you're comfortable with the signs of trigonometric functions in different quadrants, as this impacts the result when dealing with specific angle measurements.
\[ \cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B \]
When confronted with an expression like \( \cos 120^\circ \cos 30^\circ + \sin 120^\circ \sin 30^\circ \), you can recognize it as conforming to this identity, with \( A = 120^\circ \) and \( B = 30^\circ \).
By applying the identity, you transform the original compound expression into a simple function: \( \cos(120^\circ - 30^\circ) \) or \( \cos 90^\circ \). This simplification makes it much easier to evaluate the expression, as you only need to find the cosine of a single angle.
There are a few common mistakes to avoid when using this identity. Firstly, ensure that you use the correct signs – note that after the cosine, it's a plus sign connecting the two products. Secondly, make sure you're comfortable with the signs of trigonometric functions in different quadrants, as this impacts the result when dealing with specific angle measurements.
Trigonometric Functions
Trigonometric functions are the backbone of trigonometry and are essential in fields ranging from physics to engineering. These functions, including sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}), relate the angles of a triangle to the lengths of its sides in a right-angled triangle. However, they also have broader applications in modeling periodic phenomena represented on the unit circle.
Each trigonometric function has an associated graph which is periodic and repeats its values in a regular cycle, making them immensely useful in analyzing wave patterns, such as sound and light. In the context of the unit circle, these functions represent different coordinates of a point on the circle's circumference. For example, \( \cos \theta \) gives the x-coordinate, while \( \sin \theta \) gives the y-coordinate of that point.
To avoid common pitfalls when dealing with trigonometric functions, always remember: the domain (input) of these functions is an angle (commonly measured in degrees or radians), and the range (output) is a real number, which is the ratio of sides for \text{sin}, \text{cos}, and \text{tan}, or the length of various other line segments in the unit circle interpretation.
Each trigonometric function has an associated graph which is periodic and repeats its values in a regular cycle, making them immensely useful in analyzing wave patterns, such as sound and light. In the context of the unit circle, these functions represent different coordinates of a point on the circle's circumference. For example, \( \cos \theta \) gives the x-coordinate, while \( \sin \theta \) gives the y-coordinate of that point.
To avoid common pitfalls when dealing with trigonometric functions, always remember: the domain (input) of these functions is an angle (commonly measured in degrees or radians), and the range (output) is a real number, which is the ratio of sides for \text{sin}, \text{cos}, and \text{tan}, or the length of various other line segments in the unit circle interpretation.
Unit Circle
The unit circle is a powerful tool in trigonometry, defined as a circle with a radius of one unit centered at the origin of a coordinate plane. It provides a visual representation for understanding trigonometric functions at any angle. On this circle, every point corresponds to an angle in standard position (where the vertex is at the origin, one ray is on the positive x-axis, and the other ray extends in a counter-clockwise direction).
Crucially, the coordinates of any point on the unit circle can tell us the values of \( \cos \) and \( \sin \) for that angle. For a given angle \( \theta \) in standard position, where the angle's terminal side intersects the unit circle at point P, then the x-coordinate of P represents \( \cos \theta \) and the y-coordinate represents \( \sin \theta \).
\( \cos 90^\circ = 0 \) is an excellent example of applying the unit circle concept. Since 90 degrees or \( \pi/2 \) radians corresponds to the topmost point of the circle, the x-coordinate at this point is 0, meaning the cosine of 90 degrees is also 0.
To fully benefit from the unit circle, students should memorize the coordinates of common angles (like 30, 45, 60, and 90 degrees) and understand that the circle reflects the periodic nature of trigonometric functions. With this knowledge, they can easily solve a wide range of problems involving angles and their trigonometric functions.
Crucially, the coordinates of any point on the unit circle can tell us the values of \( \cos \) and \( \sin \) for that angle. For a given angle \( \theta \) in standard position, where the angle's terminal side intersects the unit circle at point P, then the x-coordinate of P represents \( \cos \theta \) and the y-coordinate represents \( \sin \theta \).
\( \cos 90^\circ = 0 \) is an excellent example of applying the unit circle concept. Since 90 degrees or \( \pi/2 \) radians corresponds to the topmost point of the circle, the x-coordinate at this point is 0, meaning the cosine of 90 degrees is also 0.
To fully benefit from the unit circle, students should memorize the coordinates of common angles (like 30, 45, 60, and 90 degrees) and understand that the circle reflects the periodic nature of trigonometric functions. With this knowledge, they can easily solve a wide range of problems involving angles and their trigonometric functions.
Other exercises in this chapter
Problem 39
In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \tan(-x) \cos x \)
View solution Problem 40
In Exercises 37-42, find the exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) using the double-angle formulas. \( \cot u = \sqrt{2}, \pi
View solution Problem 40
In Exercises 39-44, solve the multiple-angle equation. \( \sin 2x = -\dfrac{\sqrt{3}}{2} \)
View solution Problem 40
In Exercises 9-50, verify the identity \( \dfrac{\cos x - \cos y}{\sin x + \sin y} + \dfrac{\sin x - \sin y}{\cos x + \cos y} = 0 \)
View solution