Simplify Tan 480° ⋅ Sin 300° ⋅ Cos 14° ⋅ Sin (-135°) Over Sin 104° ⋅ Cos 225° Without A Calculator

Hey guys! Today, we're going to dive into simplifying a trigonometric expression without reaching for that calculator. It might seem daunting at first, but we'll break it down step-by-step, making it super easy to follow. So, let’s get started!

The Trigonometric Challenge

Our mission, should we choose to accept it, is to simplify this expression:

$$\frac{\tan 480^{\circ} \cdot \sin 300^{\circ} \cdot \cos 14^{\circ} \cdot \sin (-135^{\circ})}{\sin 104^{\circ} \cdot \cos 225^{\circ}}$$

Don't worry if it looks like a jumble of angles and trig functions right now. We'll untangle it all. To begin with simplifying trigonometric expressions, remember that key trigonometric identities and angle reduction formulas are our best friends. These tools allow us to bring down those large angles into more manageable ones and convert trig functions into their simplest forms. By understanding these concepts thoroughly, you can tackle any such problem with ease and confidence. Now, let’s see how we can apply these to our expression.

Breaking Down the Expression

First, we'll take each term individually and simplify it. This involves using the properties of trigonometric functions and reducing angles to their reference angles. Remember, angles can be coterminal (ending at the same spot after one or more full rotations) or can be negative, but we can always find an equivalent positive acute angle.

Simplifying ${\tan 480^{\circ}}$

When dealing with tangent of 480 degrees, recognize that angles repeat every 360 degrees. So, we subtract 360 from 480 to find a coterminal angle:

$$480^{\circ} - 360^{\circ} = 120^{\circ}$$

Now, we need to find ${\tan 120^{\circ}}$. Think about the unit circle. 120 degrees is in the second quadrant, where tangent is negative. The reference angle (the angle formed with the x-axis) is:

$$180^{\circ} - 120^{\circ} = 60^{\circ}$$

We know that ${\tan 60^{\circ} = \sqrt{3}}$, so:

$$\tan 120^{\circ} = -\sqrt{3}$$

Therefore:

$$\tan 480^{\circ} = -\sqrt{3}$$

Simplifying ${\sin 300^{\circ}}$

Next up, let's tackle sine of 300 degrees. This angle is in the fourth quadrant, where sine is negative. The reference angle is:

$$360^{\circ} - 300^{\circ} = 60^{\circ}$$

We know that ${\sin 60^{\circ} = \frac{\sqrt{3}}{2}}$, so:

$$\sin 300^{\circ} = -\frac{\sqrt{3}}{2}$$

Simplifying ${\cos 14^{\circ}}$

Okay, cosine of 14 degrees looks friendly enough for now. We'll keep it as is since it's already an acute angle, and we don't have any immediate simplifications. Sometimes, you’ll find that terms like these will cancel out later in the problem. So, let’s hold onto it and see what happens.

Simplifying ${\sin (-135^{\circ})}$

Now, let's deal with sine of -135 degrees. The negative angle might look tricky, but we can think of it as a clockwise rotation. This angle lands in the third quadrant, where sine is negative. To find the reference angle, we add 360 degrees to get a positive coterminal angle:

$$-135^{\circ} + 360^{\circ} = 225^{\circ}$$

Then, the reference angle (with respect to the x-axis) is:

$$225^{\circ} - 180^{\circ} = 45^{\circ}$$

We know that ${\sin 45^{\circ} = \frac{\sqrt{2}}{2}}$, so:

$$\sin (-135^{\circ}) = -\frac{\sqrt{2}}{2}$$

Simplifying ${\sin 104^{\circ}}$

Here comes sine of 104 degrees. This angle is in the second quadrant, where sine is positive. We can relate it to its supplementary angle (the angle that adds up to 180 degrees):

$$180^{\circ} - 104^{\circ} = 76^{\circ}$$

Using the identity ${\sin(180^{\circ} - x) = \sin x}$, we have:

$$\sin 104^{\circ} = \sin 76^{\circ}$$

But wait! We can also express 76 degrees as:

$$76^{\circ} = 90^{\circ} - 14^{\circ}$$

Using the identity ${\sin(90^{\circ} - x) = \cos x}$, we get:

$$\sin 104^{\circ} = \cos 14^{\circ}$$

Look at that! A sneaky way to bring in ${\cos 14^{\circ}}$, which we saw earlier. This is a classic example of how trigonometric problems often have hidden connections that simplify things beautifully.

Simplifying ${\cos 225^{\circ}}$

Lastly, let's simplify cosine of 225 degrees. This angle is in the third quadrant, where cosine is negative. The reference angle is:

$$225^{\circ} - 180^{\circ} = 45^{\circ}$$

We know that ${\cos 45^{\circ} = \frac{\sqrt{2}}{2}}$, so:

$$\cos 225^{\circ} = -\frac{\sqrt{2}}{2}$$

Putting It All Together

Now that we've simplified each term, let’s plug them back into the original expression:

$$\frac{\tan 480^{\circ} \cdot \sin 300^{\circ} \cdot \cos 14^{\circ} \cdot \sin (-135^{\circ})}{\sin 104^{\circ} \cdot \cos 225^{\circ}} = \frac{(-{\sqrt{3}}) \cdot (-\frac{{\sqrt{3}}}{2}) \cdot \cos 14^{\circ} \cdot (-\frac{{\sqrt{2}}}{2})}{\cos 14^{\circ} \cdot (-\frac{{\sqrt{2}}}{2})}$$

Time for some cancellation magic! Notice that ${\cos 14^{\circ}}$ appears in both the numerator and the denominator, so they cancel out. Also, ${-\frac{{\sqrt{2}}}{2}}$ is in both places too, so goodbye to those!

$$\frac{(-{\sqrt{3}}) \cdot (-\frac{{\sqrt{3}}}{2}) \cdot \cancel{\cos 14^{\circ}} \cdot (-\cancel{\frac{{\sqrt{2}}}{2}})}{\cancel{\cos 14^{\circ}} \cdot (-\cancel{\frac{{\sqrt{2}}}{2}})} = -{\sqrt{3}} \cdot -\frac{{\sqrt{3}}}{2}$$

Now, just multiply the remaining terms:

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

Final Simplified Answer

So, the simplified form of the original expression is:

$$\frac{3}{2}$$

And there you have it! We took a seemingly complicated trigonometric expression and, without a calculator, broke it down into a simple fraction. Isn't that satisfying? This exercise shows how understanding trigonometric identities, reference angles, and quadrant rules can empower you to solve these problems efficiently. Keep practicing, and you'll become a trig wizard in no time! Remember, mastering trigonometric simplifications involves knowing your identities, understanding angle relationships, and practicing regularly. With these skills, you’ll be able to tackle even the most intimidating trigonometric expressions.

Key Takeaways

Mastering Trigonometric Simplifications

When simplifying trigonometric equations, it's crucial to systematically reduce the angles and apply relevant identities. Start by reducing angles greater than 360 degrees or negative angles to their coterminal angles within the range of 0 to 360 degrees. This often involves adding or subtracting multiples of 360 degrees until you get an angle within this range. Next, identify the quadrant in which the angle lies. Knowing the quadrant helps determine the sign of the trigonometric function (positive or negative) because each trigonometric function has specific signs in each quadrant.

Utilizing Reference Angles

Finding the reference angle is a critical step in simplifying trigonometric expressions. The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. For angles in the second quadrant (90° < θ < 180°), the reference angle is calculated as 180° - θ. For angles in the third quadrant (180° < θ < 270°), it’s θ - 180°, and for angles in the fourth quadrant (270° < θ < 360°), it’s 360° - θ. Once you have the reference angle, you can express the trigonometric function of the original angle in terms of the trigonometric function of its reference angle, adjusting for the sign based on the quadrant.

Applying Trigonometric Identities

Key trigonometric identities are essential tools for simplifying expressions. For instance, the Pythagorean identities (${\sin^2 θ + \cos^2 θ = 1}$, ${1 + \tan^2 θ = \sec^2 θ}$, and ${1 + \cot^2 θ = \csc^2 θ}$) can help you convert between sine, cosine, tangent, and their reciprocals. The sum and difference formulas (e.g., ${\sin(A ± B)}$, ${\cos(A ± B)}$) and double-angle formulas (e.g., ${\sin(2θ)}$, ${\cos(2θ)}$) are also invaluable for breaking down complex expressions into simpler terms. Recognizing when and how to apply these identities can significantly streamline the simplification process.

Recognizing and Cancelling Common Factors

One of the most effective techniques in simplifying trigonometric expressions is recognizing and canceling common factors. This often becomes apparent after you have reduced the angles and applied trigonometric identities. Look for terms that appear in both the numerator and the denominator of an expression. Cancelling these common factors can significantly simplify the expression, making it easier to evaluate or further manipulate. In our example, we saw how ${\cos 14^{\circ}}$ and ${-\frac{{\sqrt{2}}}{2}}$ canceled out, which greatly simplified the final calculation.

Practice Makes Perfect

Like any mathematical skill, mastering trigonometric simplification requires practice. Work through a variety of problems, starting with simpler ones and gradually tackling more complex expressions. Each problem you solve will help you build familiarity with trigonometric identities, angle relationships, and simplification techniques. Keep a reference sheet of important identities and formulas handy, and don't be afraid to refer to it as needed. Over time, you'll develop an intuitive sense of which identities to apply and how to approach different types of problems.

By following these strategies and practicing regularly, you’ll enhance your ability to simplify trigonometric expressions efficiently and accurately. Remember, each step you take towards mastering these concepts builds a strong foundation for more advanced mathematical topics. Keep up the great work, and you’ll be simplifying trigonometric expressions like a pro in no time!

Let's Keep the Conversation Going!

I hope this breakdown was helpful, guys! If you have any questions or other trig expressions you'd like to simplify, drop them in the comments below. Let's keep learning and simplifying together!