Trigonometric Expression Calculation: Step-by-Step Solution

Oct 10, 2025 by ADMIN 60 views

Let's break down how to solve the trigonometric expression: $\frac{sin 315°}{tan 135°} + \frac{cot 300°}{cos 300°}$ . This involves understanding the values of sine, tangent, cotangent, and cosine at specific angles, and then performing the arithmetic. Grab your calculators (or your unit circle!), and let’s dive in!

Understanding the Trigonometric Values

Before we can evaluate the expression, we need to determine the values of each trigonometric function at the given angles. We'll use the unit circle and reference angles to find these values. Remember, the unit circle is your friend! It helps visualize these values and makes everything much easier. Trigonometry can seem daunting, but with a solid understanding of the basics, it becomes much more manageable. Understanding trigonometric values is key to unlocking more complex problems. A solid understanding of these values allows for more efficient and accurate problem-solving.

Sine of 315°

The angle 315° lies in the fourth quadrant. The reference angle is 360° - 315° = 45°. In the fourth quadrant, sine is negative. Therefore, sin 315° = -sin 45° = - $\frac{\sqrt{2}}{2}$ . When dealing with sine values, always consider the quadrant to determine the sign. The fourth quadrant, where 315° resides, dictates a negative sine value. So, we find that $sin(315°) = -\frac{\sqrt{2}}{2}$ .

Tangent of 135°

The angle 135° lies in the second quadrant. The reference angle is 180° - 135° = 45°. In the second quadrant, tangent is negative. Therefore, tan 135° = -tan 45° = -1. Don't forget that tangent values are negative in the second quadrant. So, $tan(135°) = -1$ . To find the tangent of 135°, recognize it's in the second quadrant where tangent is negative. The reference angle is 45°, and $tan(45°) = 1$ , so $tan(135°) = -1$ .

Cotangent of 300°

The angle 300° lies in the fourth quadrant. The reference angle is 360° - 300° = 60°. Cotangent is the reciprocal of tangent, and in the fourth quadrant, cotangent is negative. Therefore, cot 300° = -cot 60° = - $\frac{1}{\sqrt{3}}$ = - $\frac{\sqrt{3}}{3}$ . Cotangent values also change signs based on the quadrant. Remember that cotangent is the reciprocal of tangent. Since tangent is negative in the fourth quadrant, so is cotangent. Thus, $cot(300°) = -\frac{\sqrt{3}}{3}$ .

Cosine of 300°

The angle 300° lies in the fourth quadrant. The reference angle is 360° - 300° = 60°. In the fourth quadrant, cosine is positive. Therefore, cos 300° = cos 60° = $\frac{1}{2}$ . When evaluating cosine values, remember that cosine is positive in the fourth quadrant. Therefore, $cos(300°) = \frac{1}{2}$ .

Evaluating the Expression

Now that we have all the trigonometric values, we can substitute them into the expression and simplify:

$\frac{sin 315°}{tan 135°} + \frac{cot 300°}{cos 300°}$ = $\frac{-\frac{\sqrt{2}}{2}}{-1} + \frac{-\frac{\sqrt{3}}{3}}{\frac{1}{2}}$

Simplify the first term:

$\frac{-\frac{\sqrt{2}}{2}}{-1}$ = $\frac{\sqrt{2}}{2}$

Simplify the second term:

$\frac{-\frac{\sqrt{3}}{3}}{\frac{1}{2}}$ = - $\frac{\sqrt{3}}{3}$ * 2 = - $\frac{2\sqrt{3}}{3}$

Now, add the two terms together:

$\frac{\sqrt{2}}{2} - \frac{2\sqrt{3}}{3}$

To combine these terms, we need a common denominator, which is 6:

$\frac{3\sqrt{2}}{6} - \frac{4\sqrt{3}}{6}$ = $\frac{3\sqrt{2} - 4\sqrt{3}}{6}$

Final Answer

Therefore, the result of the expression is:

$\frac{3\sqrt{2} - 4\sqrt{3}}{6}$

So, by carefully evaluating each trigonometric function and simplifying the expression, we arrive at our final answer. Remember to always double-check your work and ensure you have the correct signs for each quadrant. This step-by-step approach makes tackling trigonometric problems much easier and more efficient. Guys, remember that practice makes perfect!

Key Takeaways:

Always identify the quadrant to determine the sign of the trigonometric function.
Use reference angles to find the values of trigonometric functions.
Simplify each term carefully before combining them.

By following these steps, you can confidently solve similar trigonometric expressions. Keep practicing, and you'll become a pro in no time!

Practice Problems:

To solidify your understanding, try these practice problems:

Evaluate: $\frac{cos 225°}{sin 330°} + \frac{tan 120°}{cot 240°}$
Simplify: $sin 150° * cos 315° - tan 225°$

Good luck, and happy calculating! Understanding trigonometric functions can unlock many complex problems. Make sure to practice and solve many problems to improve your skills.

Further Exploration

To deepen your understanding of trigonometry, consider exploring these topics:

Unit Circle: A comprehensive understanding of the unit circle is essential for mastering trigonometric functions. Familiarize yourself with the coordinates of points on the unit circle at various angles.
Trigonometric Identities: Learning trigonometric identities can simplify complex expressions and solve equations more efficiently. Some key identities include the Pythagorean identities, sum and difference identities, and double-angle identities.
Graphs of Trigonometric Functions: Understanding the graphs of sine, cosine, tangent, and other trigonometric functions can provide valuable insights into their behavior and properties.
Applications of Trigonometry: Explore real-world applications of trigonometry in fields such as physics, engineering, and navigation.

By delving deeper into these areas, you can gain a more profound appreciation for the power and versatility of trigonometry. Keep exploring, and don't be afraid to tackle challenging problems!

Tips for Success

Here are some additional tips to help you succeed in trigonometry:

Memorize Key Values: Memorize the values of sine, cosine, and tangent for common angles such as 0°, 30°, 45°, 60°, and 90°. This will save you time and effort when solving problems.
Practice Regularly: Consistent practice is crucial for mastering trigonometry. Work through a variety of problems to reinforce your understanding and develop your problem-solving skills.
Seek Help When Needed: Don't hesitate to ask for help from your teacher, classmates, or online resources if you're struggling with a particular concept or problem.
Use Visual Aids: Use visual aids such as the unit circle, graphs, and diagrams to help you visualize trigonometric functions and their relationships.
Check Your Work: Always double-check your work to ensure you haven't made any mistakes. Pay close attention to signs, units, and calculations.

By following these tips, you can improve your understanding of trigonometry and achieve success in your studies. Remember that consistent practice is the key to mastering trigonometry. It is crucial to understand all the underlying concepts.

Common Mistakes to Avoid

When working with trigonometric expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Incorrect Signs: Forgetting to consider the quadrant when determining the sign of a trigonometric function.
Using the Wrong Reference Angle: Calculating the reference angle incorrectly.
Mixing Up Trigonometric Identities: Applying trigonometric identities incorrectly.
Making Arithmetic Errors: Making mistakes in calculations, especially when dealing with fractions and radicals.
Not Simplifying Completely: Failing to simplify the expression to its simplest form.

By being aware of these common mistakes, you can avoid them and improve your accuracy when solving trigonometric problems. Always take your time and double-check your work to minimize errors. Being mindful of these common mistakes can vastly improve accuracy.

Conclusion

In conclusion, evaluating trigonometric expressions requires a solid understanding of trigonometric values, reference angles, and simplification techniques. By following a step-by-step approach and avoiding common mistakes, you can confidently solve these types of problems. Remember to practice regularly and seek help when needed. With dedication and perseverance, you can master trigonometry and unlock its many applications. Keep exploring, keep learning, and keep practicing!