Solve Trig: Sin(s+t), Tan(s+t), Quadrant Of S+t

by Omar Yusuf 48 views

Hey guys! Let's dive into a cool trigonometric problem where we'll use given information to figure out some interesting things about angles and their sums. We're going to explore how to find the sine and tangent of the sum of two angles, and even pinpoint which quadrant that sum lands in. It's like a trigonometric puzzle, and we're about to solve it!

Problem Overview

We're given that cos s = -3/5 and cos t = -8/17, and we know that both angles s and t are chilling in the second quadrant. Our mission, should we choose to accept it (and we totally do!), is threefold:

(a) Calculate sin(s + t). (b) Determine tan(s + t). (c) Identify the quadrant in which the angle s + t lies.

This problem is a fantastic journey through the world of trigonometric identities and angle relationships. We'll need to dust off our knowledge of the Pythagorean identity, sum and difference formulas, and the unit circle to conquer this challenge. So, let's get started!

Step 1: Finding sin s and sin t

The heart of this trigonometric quest lies in our ability to find sin s and sin t. We are given the cosines, but to unlock sin(s + t), we need both sines and cosines. How do we bridge this gap? Enter the Pythagorean identity: a fundamental tool in trigonometry. It's like our trusty Swiss Army knife for trigonometric problems.

The Pythagorean identity states that for any angle θ:

sin²θ + cos²θ = 1

This identity is a direct consequence of the Pythagorean theorem applied to the unit circle. It's a cornerstone of trigonometry, and we'll use it to find our missing sine values.

Calculating sin s

We know cos s = -3/5. Let's plug this into the Pythagorean identity:

sin²s + (-3/5)² = 1 sin²s + 9/25 = 1

Now, let's isolate sin²s:

sin²s = 1 - 9/25 sin²s = 16/25

Taking the square root of both sides gives us:

sin s = ±4/5

But wait! We need to choose the correct sign. This is where the quadrant information comes to the rescue. We're told that angle s is in the second quadrant. Remember, in the second quadrant, sine is positive (think of the mnemonic "All Students Take Calculus," where "Students" indicates sine is positive in the second quadrant). Therefore:

sin s = 4/5

Calculating sin t

Now, let's repeat the process for angle t. We know cos t = -8/17. Plugging this into the Pythagorean identity:

sin²t + (-8/17)² = 1 sin²t + 64/289 = 1

Isolating sin²t:

sin²t = 1 - 64/289 sin²t = 225/289

Taking the square root:

sin t = ±15/17

Again, we need to consider the quadrant. Angle t is also in the second quadrant, where sine is positive. Therefore:

sin t = 15/17

So, we've successfully found sin s and sin t using the Pythagorean identity and the quadrant information. It's like we've gathered the necessary ingredients for our trigonometric recipe!

Step 2: Finding sin(s + t)

Now that we've determined the values of sin s, sin t, cos s, and cos t, we're ready to tackle the first part of our mission: finding sin(s + t). To do this, we'll employ the sum identity for sine. This identity is a powerful tool that allows us to express the sine of a sum of angles in terms of the sines and cosines of the individual angles. It's like having a special key that unlocks the value of sin(s + t).

The sum identity for sine is given by:

sin(s + t) = sin s cos t + cos s sin t

This formula might look a bit intimidating at first, but it's actually quite straightforward to use. We simply plug in the values we've already found for sin s, cos s, sin t, and cos t.

Let's substitute the values we calculated earlier:

sin(s + t) = (4/5)(-8/17) + (-3/5)(15/17)

Now, we perform the multiplication:

sin(s + t) = -32/85 - 45/85

Combining the fractions, we get:

sin(s + t) = -77/85

So, we've successfully calculated sin(s + t) using the sum identity for sine. It's like we've built a bridge to connect the individual angle values to the value of their sum!

Step 3: Finding tan(s + t)

Next on our trigonometric to-do list is finding tan(s + t). We have a couple of ways we could approach this, which gives us flexibility in our problem-solving strategy. We can think of this like choosing the best route to our destination.

Method 1: Using the Tangent Sum Identity

One way is to use the tangent sum identity directly. This identity expresses the tangent of a sum of angles in terms of the tangents of the individual angles. It's a specialized tool that's perfect for this job.

The tangent sum identity is given by:

tan(s + t) = (tan s + tan t) / (1 - tan s tan t)

To use this, we first need to find tan s and tan t. Remember that tangent is sine divided by cosine:

tan s = sin s / cos s = (4/5) / (-3/5) = -4/3 tan t = sin t / cos t = (15/17) / (-8/17) = -15/8

Now, we can plug these values into the tangent sum identity:

tan(s + t) = (-4/3 + (-15/8)) / (1 - (-4/3)(-15/8))

Let's simplify this expression. First, we find a common denominator for the fractions in the numerator:

tan(s + t) = ((-32/24) + (-45/24)) / (1 - (60/24)) tan(s + t) = (-77/24) / (1 - 5/2)

Now, simplify the denominator:

tan(s + t) = (-77/24) / (-3/2)

Dividing fractions is the same as multiplying by the reciprocal:

tan(s + t) = (-77/24) * (-2/3) tan(s + t) = 77/36

Method 2: Using sin(s + t) and cos(s + t)

Another way to find tan(s + t) is to use the fact that tangent is sine divided by cosine:

tan(s + t) = sin(s + t) / cos(s + t)

We already know sin(s + t) = -77/85. To use this method, we need to find cos(s + t). We can use the cosine sum identity:

cos(s + t) = cos s cos t - sin s sin t

Plugging in the values we know:

cos(s + t) = (-3/5)(-8/17) - (4/5)(15/17) cos(s + t) = 24/85 - 60/85 cos(s + t) = -36/85

Now we can find tan(s + t):

tan(s + t) = sin(s + t) / cos(s + t) = (-77/85) / (-36/85) tan(s + t) = 77/36

Both methods lead us to the same answer:

tan(s + t) = 77/36

We've successfully navigated the world of tangent sum identities and found our target value. It's like we've unlocked a secret passage in our trigonometric adventure!

Step 4: Determining the Quadrant of s + t

Our final challenge is to determine the quadrant in which the angle s + t lies. We've already calculated sin(s + t) = -77/85 and tan(s + t) = 77/36. We also found that cos(s + t) = -36/85 in the alternative method for finding tan(s+t). This is like putting the final pieces of a puzzle together to reveal the complete picture.

To figure out the quadrant, let's analyze the signs of sine, cosine, and tangent:

  • sin(s + t) = -77/85 (negative)
  • cos(s + t) = -36/85 (negative)
  • tan(s + t) = 77/36 (positive)

Remember our quadrant sign chart:

  • Quadrant I: All trigonometric functions are positive.
  • Quadrant II: Sine is positive, others are negative.
  • Quadrant III: Tangent is positive, others are negative.
  • Quadrant IV: Cosine is positive, others are negative.

We see that sin(s + t) and cos(s + t) are negative, while tan(s + t) is positive. This combination of signs points us directly to:

Quadrant III

Therefore, the angle s + t lies in the third quadrant. We've successfully pinpointed the location of our combined angle, completing our trigonometric quest!

Conclusion

Woah, guys! We've successfully navigated this trigonometric adventure! We started with cos s and cos t, ventured through the Pythagorean identity to find sin s and sin t, unlocked the secrets of the sine and tangent sum identities to calculate sin(s + t) and tan(s + t), and finally, used the signs of our trigonometric functions to determine that s + t lies in the third quadrant.

This problem beautifully illustrates the interconnectedness of trigonometric concepts and the power of identities. It's like we've used a map of trigonometric relationships to guide us to our final destination. Keep practicing, and you'll become a trigonometric wizard in no time!