Simplifying Cos 6° Cos 11° - Sin 6° Sin 11° As A Trigonometric Function
In the realm of trigonometry, simplifying complex expressions into more manageable forms is a fundamental skill. This often involves using trigonometric identities to rewrite expressions as functions of a single angle. Our focus here is to dissect the expression cos 6° cos 11° - sin 6° sin 11° and transform it into a trigonometric function of a single angle, and then evaluate the result. This task leverages the cosine angle addition identity, a cornerstone of trigonometric manipulations. We will walk through the process step by step, making it easy to understand and apply these concepts to similar problems.
Before diving into the specific problem, let's briefly touch on trigonometric identities. Trigonometric identities are equations that are true for all values of the variables involved. They are essential tools for simplifying trigonometric expressions and solving trigonometric equations. Among these identities, the cosine angle addition and subtraction identities are particularly powerful. The cosine angle addition identity states that:
cos(A + B) = cos A cos B - sin A sin B
Similarly, the cosine angle subtraction identity is:
cos(A - B) = cos A cos B + sin A sin B
These identities allow us to combine or separate angles within cosine functions, which is precisely what we need for our problem. Recognizing patterns that match these identities is key to simplifying trigonometric expressions effectively. By mastering these identities, you gain a significant advantage in manipulating and solving a wide range of trigonometric problems. In the following sections, we will demonstrate how to apply the cosine angle addition identity to the given expression.
The expression we are given is cos 6° cos 11° - sin 6° sin 11°. Notice how this expression closely resembles the right-hand side of the cosine angle addition identity:
cos(A + B) = cos A cos B - sin A sin B
By carefully comparing the given expression with the identity, we can identify the corresponding angles. Here, A = 6° and B = 11°. Substituting these values into the cosine angle addition identity, we can rewrite the expression as:
cos(6° + 11°) = cos 6° cos 11° - sin 6° sin 11°
This step is crucial as it transforms the original expression, which involves multiple trigonometric functions, into a single cosine function with a combined angle. This simplification makes the expression much easier to evaluate and understand. The ability to recognize and apply these identities is a fundamental skill in trigonometry, enabling us to manipulate and simplify complex expressions into more manageable forms. In the next step, we will further simplify the expression by adding the angles and then evaluating the cosine of the resulting angle.
After rewriting the expression using the cosine angle addition identity, we have:
cos(6° + 11°)
The next step is to simplify the angle inside the cosine function. Adding the angles 6° and 11° gives us:
6° + 11° = 17°
So, our expression now becomes:
cos(17°)
This is the trigonometric function of a single angle that we were aiming for. To evaluate cos(17°), we can use a calculator. Make sure your calculator is set to degree mode. Inputting cos(17°) into a calculator, we get approximately:
cos(17°) ≈ 0.9563
Therefore, the value of the expression cos 6° cos 11° - sin 6° sin 11° is approximately 0.9563. This process demonstrates how trigonometric identities can be used to simplify complex expressions and obtain numerical values. By combining angles and using a calculator for the final evaluation, we have successfully transformed and solved the given problem. This approach highlights the power and utility of trigonometric identities in mathematical problem-solving.
In summary, the expression cos 6° cos 11° - sin 6° sin 11° can be written as a trigonometric function of a single angle using the cosine angle addition identity. By recognizing the pattern and applying the identity, we transformed the expression into cos(17°). Evaluating cos(17°) using a calculator, we found that it is approximately 0.9563. This completes the process of rewriting the expression and finding its value.
Trigonometric function of a single angle: cos(17°)
Evaluated value: ≈ 0.9563
This exercise underscores the importance of trigonometric identities in simplifying and solving trigonometric problems. By mastering these identities, you can effectively manipulate expressions and find solutions. The cosine angle addition identity, in particular, is a powerful tool for combining angles and simplifying expressions involving cosine and sine functions. Understanding and applying these concepts is crucial for success in trigonometry and related fields.
To further solidify your understanding of trigonometric identities and their applications, let's explore additional examples and practice problems. Working through a variety of problems will help you become more comfortable with recognizing patterns and applying the appropriate identities. Here are a few examples to consider:
- Simplify sin(45° + θ): Use the sine angle addition identity, which states that sin(A + B) = sin A cos B + cos A sin B. Substitute A = 45° and B = θ, and then simplify using the known values of sin(45°) and cos(45°).
- Simplify cos(75°): Recognize that 75° can be written as the sum of two known angles, such as 45° and 30°. Then, use the cosine angle addition identity to expand and simplify the expression.
- Evaluate sin(15°): Similar to the previous example, express 15° as the difference of two known angles, such as 45° and 30°. Then, use the sine angle subtraction identity to evaluate the expression.
- Rewrite sin(2θ) in terms of sin(θ) and cos(θ): Use the sine double angle identity, which is a special case of the sine angle addition identity where A = B = θ.
- Rewrite cos(2θ) in terms of cos(θ): Use the cosine double angle identity, which has multiple forms. Choose the form that expresses cos(2θ) solely in terms of cos(θ).
By tackling these problems, you will develop a stronger grasp of how to apply trigonometric identities in different scenarios. Remember, practice is key to mastering these concepts. The more you work with trigonometric identities, the more intuitive they will become.
In conclusion, the ability to manipulate and simplify trigonometric expressions is a crucial skill in mathematics. We have demonstrated how to rewrite cos 6° cos 11° - sin 6° sin 11° as a trigonometric function of a single angle using the cosine angle addition identity. This process involved recognizing the pattern, applying the identity, simplifying the angle, and evaluating the cosine function. The final result was cos(17°), which is approximately 0.9563.
Throughout this discussion, we have emphasized the importance of trigonometric identities as tools for simplification. The cosine angle addition identity, cos(A + B) = cos A cos B - sin A sin B, played a central role in our solution. We also touched on other identities, such as the cosine angle subtraction identity, sine angle addition and subtraction identities, and double angle identities, highlighting their versatility in solving various trigonometric problems.
Mastering these identities and practicing their application will significantly enhance your problem-solving abilities in trigonometry and related areas of mathematics. Remember to always look for patterns, choose the appropriate identity, and simplify step by step. With consistent practice, you will become proficient in transforming and evaluating trigonometric expressions, making complex problems more manageable and understandable. Trigonometry is a fundamental aspect of mathematics, and the skills you develop in this area will serve you well in further studies and applications.