Signs Of Trigonometric Expressions: A Detailed Analysis

by ADMIN 56 views

Let's dive into the fascinating world of trigonometry and figure out the signs of some tricky expressions! This article is all about understanding how the values of sine, cosine, tangent, cotangent, secant, and cosecant change in different quadrants, and how these changes affect the signs of more complex expressions. We'll break it down step by step, so you'll be a pro in no time!

Determining the Signs of a, b, and c

Okay, guys, let's kick things off by figuring out the signs of a, b, and c where:

  • a = sin42° - tan53°
  • b = cot12° - cos12°
  • c = sin24° - cos24°

Analyzing a = sin42° - tan53°

So, to determine the sign of a, we need to compare sin42° and tan53°. Remember, we're dealing with angles in degrees here. To really nail this, let's think about the unit circle and the behavior of sine and tangent in the first quadrant (where angles between 0° and 90° live).

  • sin42°: Sine is positive in the first and second quadrants. Since 42° is in the first quadrant, sin42° is positive. We can estimate its value; it will be less than sin45° which is approximately 0.707.
  • tan53°: Tangent is also positive in the first quadrant. 53° is in the first quadrant, making tan53° positive. Tangent increases faster than sine in the first quadrant. Thinking about the special angles, tan45° = 1. Since 53° is greater than 45°, tan53° will be greater than 1.

Therefore, we have a positive value (sin42°) subtracting a larger positive value (tan53°). This means a is negative.

In summary, to find the sign of 'a', we analyzed that sin 42° is a positive value less than 1, and tan 53° is a positive value greater than 1. Since we are subtracting a larger positive value from a smaller one, the result 'a' is negative. This approach highlights the importance of understanding the range and behavior of trigonometric functions in different quadrants.

Figuring Out the Sign of b = cot12° - cos12°

Next up, let's tackle b = cot12° - cos12°. This one involves cotangent and cosine, both of which are positive in the first quadrant (where 12° lives). But, we need to figure out which one is bigger.

  • cot12°: Cotangent is the reciprocal of tangent (cot θ = 1/tan θ). For small angles (close to 0°), cotangent is very large and positive because tangent is very small. So, cot12° is a positive value greater than 1.
  • cos12°: Cosine is positive in the first and fourth quadrants. For angles close to 0°, cosine is close to 1. So, cos12° is a positive value close to 1.

We're subtracting a value close to 1 (cos12°) from a value greater than 1 (cot12°). This means b is positive.

To recap the sign determination for 'b', we observed that cot 12° is positive and significantly greater than 1, while cos 12° is positive and close to 1. Subtracting cos 12° from cot 12° results in a positive value. This part illustrates the importance of recognizing how trigonometric functions behave for small angles, which is crucial for comparative analysis.

Cracking the Case of c = sin24° - cos24°

Last but not least, we have c = sin24° - cos24°. Both sine and cosine are positive in the first quadrant. The key here is to remember the behavior of sine and cosine in the first quadrant. Sine increases from 0 to 1 as the angle goes from 0° to 90°, while cosine decreases from 1 to 0.

They are equal at 45 degrees (sin 45° = cos 45°). Therefore:

  • sin24°: Will be a positive value less than sin45°.
  • cos24°: Will be a positive value greater than cos45°.

Since 24° is less than 45°, sin24° is less than cos24°. We're subtracting a larger value (cos24°) from a smaller value (sin24°). This means c is negative.

To clarify the sign of 'c', we deduced that sin 24° is positive but less than cos 24°, which is also positive. This is because in the first quadrant, up to 45 degrees, cosine values are greater than sine values for the same angle. Thus, the subtraction results in a negative value. This analysis underscores the significance of knowing the comparative rates of change of sine and cosine functions.

Finding the Signs of x, y, and z

Alright, now let's switch gears and figure out the signs of x, y, and z, given:

  • x = cot1° - tan1°
  • y = cos12° - sin79°
  • z = sin11° - sec11°

Unraveling x = cot1° - tan1°

For x = cot1° - tan1°, we're dealing with cotangent and tangent of a very small angle (1°).

  • cot1°: As we discussed earlier, cotangent is the reciprocal of tangent. For very small angles, cotangent is large and positive.
  • tan1°: For very small angles, tangent is also small and positive.

So, we're subtracting a small positive value (tan1°) from a large positive value (cot1°). This means x is positive.

In summary, to determine the sign of 'x', we recognized that cot 1° is a large positive value for such a small angle, while tan 1° is a small positive value. The difference between a large positive number and a small positive number is positive. This example reinforces the concept of limits and the behavior of trigonometric functions near zero.

Deciphering y = cos12° - sin79°

Now, let's look at y = cos12° - sin79°. This one's a bit sneaky because it involves sine and cosine of different angles. But, we can use a handy trigonometric identity: sin(90° - θ) = cos θ. So, sin79° = sin(90° - 11°) = cos11°.

Now we can rewrite y as: y = cos12° - cos11°

Cosine is a decreasing function in the first quadrant. This means as the angle increases, the cosine value decreases. Since 12° is greater than 11°, cos12° is less than cos11°. We're subtracting a larger value (cos11°) from a smaller value (cos12°). This means y is negative.

To summarize, to find the sign of 'y', we converted sin 79° to cos 11° using the identity sin(90° - θ) = cos θ. Then, we used the property that cosine is a decreasing function in the first quadrant. Since 12° > 11°, cos 12° < cos 11°, and thus the subtraction yields a negative value. This step exemplifies the utility of trigonometric identities in simplifying expressions for comparison.

Cracking z = sin11° - sec11°

Finally, let's figure out the sign of z = sin11° - sec11°. Remember that secant is the reciprocal of cosine (sec θ = 1/cos θ).

  • sin11°: Sine is positive in the first quadrant. For small angles, sine is small.
  • sec11°: Cosine is positive in the first quadrant, and for angles close to 0°, cosine is close to 1. Therefore, secant (the reciprocal of cosine) will be slightly greater than 1.

So, we're subtracting a value slightly greater than 1 (sec11°) from a value less than 1 (sin11°). This means z is negative.

For the sign of 'z', we considered that sec 11° is the reciprocal of cos 11°, and since cos 11° is a value close to 1, sec 11° is slightly greater than 1. In contrast, sin 11° is positive but less than 1 for a small angle like 11°. Therefore, subtracting sec 11° from sin 11° results in a negative number. This highlights the relationship between trigonometric functions and their reciprocals.

Conclusion

Phew! We've successfully navigated through these trigonometric expressions and determined the signs of a, b, c, x, y, and z. The key takeaways here are:

  • Understanding the signs of trigonometric functions in different quadrants.
  • Knowing the behavior of sine, cosine, tangent, and their reciprocals for small angles.
  • Using trigonometric identities to simplify expressions.
  • Comparing the relative magnitudes of trigonometric functions.

Keep practicing, and you'll become a trigonometry master in no time! Remember folks, understanding the fundamentals is key to tackling more complex problems. So, keep exploring and have fun with math! We've shown here not just the answers, but the thought process and reasoning behind them. Understanding why something is true is always more valuable than just memorizing the answer. Keep that curiosity burning!