Problem 32 - Precalculus with Limits

Question

In Exercises 29 - 36, write the expression as the sine, cosine,or tangent of an angle. \( \cos 130^\circ \cos 40^\circ - \sin 130^\circ \sin 40^\circ \)

Step-by-Step Solution

Answer

The expression \( \cos 130^\circ \cos 40^\circ - \sin 130^\circ \sin 40^\circ \) can be written as \( \cos 170^\circ \)

Step 1: Identify Corresponding Angles

Here the angles are \( A = 130^\circ \) and \( B = 40^\circ \). Insert them into the Cosine of Sum formula.

Step 2: Apply the Cosine of Sum formula

According to the formula, \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \), it can be written as \( \cos(130^\circ + 40^\circ) \).

Step 3: Simplify the Expression

Simplify the addition inside the cosine function, which is \( \cos(170^\circ) \).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all included values of the variables they contain. These identities are critical for simplifying complex trigonometric expressions and solving trigonometry problems.

One of the most essential identities is the Cosine of Sum formula, represented as \( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \). It allows us to express the cosine of a sum of two angles in terms of sines and cosines of the individual angles. This particular identity is extremely useful when dealing with exercises like combining angle measures, or when the direct evaluation of an angle is complicated.

Understanding and memorizing these identities can significantly aid students in simplifying trigonometric expressions and solving equations. It's often helpful to think of them as tools in a toolkit, ready to be applied whenever they are needed in trigonometry problems.

Cosine Function

The cosine function is one of the basic trigonometric functions, often symbolized as \( \cos \). It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In terms of a unit circle, it is the x-coordinate of a point on the circle's circumference at a specific angle from the positive x-axis.

One of the key properties of the cosine function is its periodicity, meaning it repeats its values in a regular pattern along the angle measure. Importantly, the cosine function is also an even function, which implies that \( \cos(\theta) = \cos(-\theta) \). This property can be particularly helpful when manipulating expressions involving cosine or when analyzing the symmetry of trigonometric functions.

Usage in Solving Equations

When solving trigonometric equations, understanding the cosine function's characteristics enables students to predict the function's behavior, determine its zeros, and apply identities effectively to solve complex problems. It is indispensable for students to grasp these fundamental concepts to apply them in various scenarios within trigonometry and beyond.

Trigonometry in Precalculus

Trigonometry is an essential component of precalculus, building the groundwork for calculus concepts such as limits, derivatives, and integrals. In precalculus, trigonometry involves the study of the relationships between the angles and sides of triangles, particularly right-angled triangles, as well as the properties of trigonometric functions such as sine, cosine, and tangent.

Students often encounter trigonometry in precalculus through various applications, including analyzing periodic phenomena, solving real-world problems, and understanding the behavior of waves and oscillations. Mastery of trigonometric identities and functions is crucial because they are used extensively to simplify expressions and solve equations encountered in upper-level math and many science fields.

Link to Calculus

As students transition into calculus, a solid foundation in trigonometry allows them to tackle more complex problems with confidence. The sine and cosine functions, for instance, are integral in describing circular motion and modeling naturally occurring patterns, which are topics that often appear in calculus courses. This interconnectivity exemplifies why a deep understanding of trigonometry is a vital step in the journey of learning advanced mathematics.