Trigonometry Exploration Find Sin(s+t) Tan(s+t) And Quadrant Of S+t

Hey guys! Today, we're diving deep into the fascinating world of trigonometry, where we'll be using our knowledge of trigonometric identities to solve a pretty cool problem. We're given some information about the cosine of angle 's' and the sine of angle 't', and our mission is to find (a) the sine of the sum of these angles (sin(s+t)), (b) the tangent of the sum of the angles (tan(s+t)), and (c) the quadrant in which the sum of the angles (s+t) lies. Buckle up, because it's going to be an exciting ride!

The Problem at Hand

Let's break down the problem we're tackling. We know that:

• cos(s) = 12/13
• sin(t) = -4/5

Also, we know that the angles s and t satisfy certain conditions. Our ultimate goal is to figure out sin(s+t), tan(s+t), and the quadrant where s+t resides. This problem beautifully illustrates how we can leverage trigonometric identities and our understanding of the unit circle to solve complex problems. We will use the sum identities for sine and tangent, along with the Pythagorean identity, to navigate our way to the solution. Keep your thinking caps on, folks!

Part (a): Finding sin(s+t)

The Sine Sum Identity

To kick things off, we'll use the sine sum identity, which is a cornerstone in trigonometry. This identity allows us to express the sine of the sum of two angles in terms of the sines and cosines of the individual angles. The identity is as follows:

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

This formula is like a magic key that unlocks the solution. We already know cos(s) and sin(t), but we need to find sin(s) and cos(t) to plug into this equation. This is where our knowledge of the Pythagorean identity comes into play.

Finding sin(s) and cos(t) using the Pythagorean Identity

The Pythagorean identity is another fundamental concept in trigonometry, linking the sine and cosine of an angle. It's like the backbone of trigonometric relationships, and it states:

sin²(θ) + cos²(θ) = 1

Where θ (theta) is any angle. We can use this identity to find sin(s) and cos(t). Let's start with finding sin(s). We know cos(s) = 12/13. Plugging this into the Pythagorean identity, we get:

sin²(s) + (12/13)² = 1 sin²(s) + 144/169 = 1 sin²(s) = 1 - 144/169 sin²(s) = 25/169

Taking the square root of both sides, we get:

sin(s) = ±5/13

Now, this is a crucial point. We have two possible values for sin(s): 5/13 and -5/13. To determine the correct sign, we need more information about the angle 's'. Since the original problem states that s is in a specific quadrant, and the cosine is positive in that quadrant, the sine must also be positive. Therefore, we choose the positive value:

sin(s) = 5/13

Next, let's find cos(t). We know sin(t) = -4/5. Using the Pythagorean identity again:

(-4/5)² + cos²(t) = 1 16/25 + cos²(t) = 1 cos²(t) = 1 - 16/25 cos²(t) = 9/25

Taking the square root of both sides:

cos(t) = ±3/5

Similarly, we have two possible values for cos(t). Since the sine of t is negative, and t falls in a quadrant where sine is negative, the cosine must be positive in that quadrant. Thus, we choose the positive value:

cos(t) = 3/5

Plugging the Values into the Sine Sum Identity

Now that we have all the pieces of the puzzle, we can finally find sin(s+t). We know:

• sin(s) = 5/13
• cos(s) = 12/13
• sin(t) = -4/5
• cos(t) = 3/5

Plugging these values into the sine sum identity:

sin(s + t) = (5/13)(3/5) + (12/13)(-4/5) sin(s + t) = 15/65 - 48/65 sin(s + t) = -33/65

So, we've successfully found sin(s+t)!

Part (b): Finding tan(s+t)

The Tangent Sum Identity

Now, let's tackle the second part of the problem: finding tan(s+t). We'll employ the tangent sum identity, which is another crucial formula in trigonometry. This identity expresses the tangent of the sum of two angles in terms of the individual tangents. The formula is:

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

To use this identity, we need to determine tan(s) and tan(t).

Finding tan(s) and tan(t)

Remember that the tangent of an angle is defined as the ratio of its sine to its cosine:

tan(θ) = sin(θ) / cos(θ)

We already know the values of sin(s), cos(s), sin(t), and cos(t), so we can easily calculate tan(s) and tan(t).

For tan(s):

tan(s) = sin(s) / cos(s) = (5/13) / (12/13) = 5/12

For tan(t):

tan(t) = sin(t) / cos(t) = (-4/5) / (3/5) = -4/3

Plugging the Values into the Tangent Sum Identity

Now that we have tan(s) and tan(t), we can substitute these values into the tangent sum identity:

tan(s + t) = (5/12 + (-4/3)) / (1 - (5/12)(-4/3)) tan(s + t) = (5/12 - 16/12) / (1 + 20/36) tan(s + t) = (-11/12) / (56/36) tan(s + t) = (-11/12) / (14/9) tan(s + t) = (-11/12) * (9/14) tan(s + t) = -33/56

We've successfully found tan(s+t)!

Part (c): Determining the Quadrant of s+t

Analyzing the Signs of sin(s+t) and tan(s+t)

The final part of our mission is to determine which quadrant the angle (s+t) lies in. To do this, we'll analyze the signs of sin(s+t) and tan(s+t). Remember that the signs of trigonometric functions vary depending on the quadrant:

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, cosine and tangent are negative.
• Quadrant III: Tangent is positive, sine and cosine are negative.
• Quadrant IV: Cosine is positive, sine and tangent are negative.

We found that:

• sin(s + t) = -33/65 (negative)
• tan(s + t) = -33/56 (negative)

Since both sine and tangent are negative, (s+t) must lie in Quadrant IV.

Conclusion

Wow, we've really put our trigonometric skills to the test! By using the sine and tangent sum identities, along with the Pythagorean identity, we were able to find:

• (a) sin(s + t) = -33/65
• (b) tan(s + t) = -33/56
• (c) s + t lies in Quadrant IV

This problem perfectly illustrates how understanding fundamental trigonometric identities can help us solve complex problems. Keep practicing, and you'll become a trigonometry master in no time! Remember, math is a journey, not a destination. So, keep exploring, keep learning, and most importantly, keep having fun!