Finding Sin Θ When Cos Θ Is Negative And Tan Θ Is Positive

by Pedro Alvarez 59 views

Hey guys! Today, we're diving into a trigonometry problem where we need to find the value of $\sin \theta$ given some clues about $\cos \theta$ and $ an \theta$. It might sound a bit intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand.

Understanding the Problem

So, the problem tells us that $\cos \theta = -\frac{2}{5}$ and $\tan \theta > 0$. Our mission, should we choose to accept it (and we do!), is to figure out what $\sin \theta$ is. To nail this, we need to dust off our knowledge of trigonometric functions, their signs in different quadrants, and a handy-dandy Pythagorean identity.

Decoding the Clues

First, let's look at $\cos \theta = -\frac{2}{5}$. Remember, cosine is negative in the second and third quadrants. This is a crucial piece of information because it narrows down where our angle $\ heta$ could possibly be located on the unit circle. Think of the unit circle like a map for angles and their trig values. Knowing the sign of cosine helps us pinpoint the right neighborhood.

Next up, we have $\tan \theta > 0$. Tangent is positive in the first and third quadrants. This is because tangent is the ratio of sine to cosine, and a positive result means either both sine and cosine are positive (first quadrant) or both are negative (third quadrant).

Putting the Pieces Together

Now comes the detective work! We know $\cos \theta$ is negative (second and third quadrants) and $\tan \theta$ is positive (first and third quadrants). The only quadrant that satisfies both conditions is the third quadrant. This is where the magic happens! Knowing the quadrant is super important because it tells us the sign of $\sin \theta$. In the third quadrant, both sine and cosine are negative.

Using the Pythagorean Identity

Alright, now that we know which quadrant we're in, we can use the Pythagorean identity to find $\sin \theta$. This identity is a cornerstone of trigonometry, and it states: $\sin^2 \theta + \cos^2 \theta = 1$. It's like the Pythagorean theorem but for trig functions, and it's incredibly useful.

Plugging in the Values

We already know $\cos \theta = -\frac{2}{5}$, so let's plug that into the identity:

sin⁡2θ+(−25)2=1\sin^2 \theta + \left(-\frac{2}{5}\right)^2 = 1

Let's simplify this:

sin⁡2θ+425=1\sin^2 \theta + \frac{4}{25} = 1

Solving for $\sin^2 \theta$

To isolate $\sin^2 \theta$, we subtract $\frac{4}{25}$ from both sides:

sin⁡2θ=1−425\sin^2 \theta = 1 - \frac{4}{25}

sin⁡2θ=2525−425\sin^2 \theta = \frac{25}{25} - \frac{4}{25}

sin⁡2θ=2125\sin^2 \theta = \frac{21}{25}

Finding $\sin \theta$

Now, we need to take the square root of both sides to find $\sin \theta$:

sin⁡θ=±2125\sin \theta = \pm\sqrt{\frac{21}{25}}

sin⁡θ=±215\sin \theta = \pm\frac{\sqrt{21}}{5}

Choosing the Correct Sign

Remember, we determined that $\ heta$ is in the third quadrant, where sine is negative. So, we choose the negative root:

sin⁡θ=−215\sin \theta = -\frac{\sqrt{21}}{5}

Final Answer

And there you have it! The value of $\sin \theta$ is $-\frac{\sqrt{21}}{5}$. We solved this by understanding the signs of trigonometric functions in different quadrants and using the Pythagorean identity. It's like a puzzle, and we just put all the pieces together!

Breaking Down the Quadrants and Trigonometric Signs

To really nail these types of problems, let's take a closer look at how trigonometric functions behave in different quadrants. This is like knowing the rules of the road for navigating the unit circle. Understanding these rules will make figuring out the signs of sine, cosine, and tangent a piece of cake.

The Unit Circle: Your Trigonometric Map

Think of the unit circle as your map for all things trigonometry. It's a circle with a radius of 1, centered at the origin (0,0) of the coordinate plane. Angles are measured counterclockwise from the positive x-axis. Each point on the circle corresponds to an angle, and the coordinates of that point are directly related to the cosine and sine of that angle.

The x-coordinate of a point on the unit circle is the cosine of the angle, and the y-coordinate is the sine of the angle. This is a fundamental concept, so make sure you've got it down! Tangent, as we mentioned earlier, is the ratio of sine to cosine (y/x).

Quadrant I: The Land of Positivity

In the first quadrant (0° to 90°), everything is positive. Sine, cosine, and tangent are all greater than zero. This is because both the x and y coordinates are positive in this quadrant. It's like the happy zone of the unit circle!

  • Sine (sin θ): Positive (y-coordinate is positive)
  • Cosine (cos θ): Positive (x-coordinate is positive)
  • Tangent (tan θ): Positive (y/x, positive divided by positive)

Quadrant II: Sine's Domain

In the second quadrant (90° to 180°), sine is positive, but cosine and tangent are negative. Here, the y-coordinate is positive, but the x-coordinate is negative. So, sine gets the spotlight in this quadrant.

  • Sine (sin θ): Positive (y-coordinate is positive)
  • Cosine (cos θ): Negative (x-coordinate is negative)
  • Tangent (tan θ): Negative (y/x, positive divided by negative)

Quadrant III: Tangent's Territory

The third quadrant (180° to 270°) is where tangent reigns supreme. Both sine and cosine are negative in this quadrant, which means their ratio, tangent, is positive. It's like the underdog quadrant where tangent finally gets its due.

  • Sine (sin θ): Negative (y-coordinate is negative)
  • Cosine (cos θ): Negative (x-coordinate is negative)
  • Tangent (tan θ): Positive (y/x, negative divided by negative)

Quadrant IV: Cosine's Corner

Finally, we have the fourth quadrant (270° to 360°), where cosine is positive, and sine and tangent are negative. The x-coordinate is positive, while the y-coordinate is negative. Cosine gets to shine in this corner of the unit circle.

  • Sine (sin θ): Negative (y-coordinate is negative)
  • Cosine (cos θ): Positive (x-coordinate is positive)
  • Tangent (tan θ): Negative (y/x, negative divided by positive)

Mnemonic Devices to the Rescue!

Memorizing these quadrant rules can be a bit tricky, but don't worry, there are some handy mnemonic devices to help you out. One popular one is "All Students Take Calculus":

  • All: All trigonometric functions are positive in Quadrant I.
  • Students: Sine is positive in Quadrant II.
  • Take: Tangent is positive in Quadrant III.
  • Calculus: Cosine is positive in Quadrant IV.

Another one is "ASTC" (All, Sine, Tangent, Cosine), which represents the positive trigonometric function in each quadrant, starting from Quadrant I and going counterclockwise.

Why This Matters

Understanding these quadrant rules is crucial for solving trigonometric problems because it helps you determine the sign of the trigonometric functions. This is especially important when you're given information about one trigonometric function and need to find the others, like in our original problem. Knowing the quadrant narrows down the possibilities and allows you to choose the correct sign for your answer.

The Power of the Pythagorean Identity

Now that we've mastered the quadrants, let's delve deeper into the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. This identity is not just a formula; it's a fundamental relationship between sine and cosine that stems directly from the Pythagorean theorem. Think of it as the glue that holds sine and cosine together.

From Pythagorean Theorem to Trigonometric Identity

To understand where this identity comes from, let's revisit the unit circle. Imagine a point (x, y) on the unit circle corresponding to an angle $\ heta$. The x-coordinate is $\ ext{cos } \theta$, and the y-coordinate is $\ ext{sin } \theta$. If we draw a right triangle with the x-axis, the y-axis, and the line from the origin to the point (x, y), we have a triangle with legs of length |x| and |y| and a hypotenuse of length 1 (since it's the radius of the unit circle).

The Pythagorean theorem tells us that $a^2 + b^2 = c^2$, where a and b are the lengths of the legs and c is the length of the hypotenuse. In our case, this translates to:

x2+y2=12x^2 + y^2 = 1^2

Since $x = \cos \theta$ and $y = \sin \theta$, we can substitute these into the equation:

(cos⁡θ)2+(sin⁡θ)2=1(\cos \theta)^2 + (\sin \theta)^2 = 1

And that's how we arrive at the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. It's a direct consequence of the Pythagorean theorem applied to the unit circle.

Using the Identity to Find Missing Trigonometric Values

The Pythagorean identity is a powerful tool because it allows us to find the value of $\ ext{sin } \theta$ if we know $\ ext{cos } \theta$, or vice versa. It's like having a secret weapon in your trigonometry arsenal.

For example, if we know $\ ext{cos } \theta$, we can rearrange the identity to solve for $\ ext{sin } \theta$:

sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta

sin⁡θ=±1−cos⁡2θ\sin \theta = \pm\sqrt{1 - \cos^2 \theta}

Similarly, if we know $\ ext{sin } \theta$, we can solve for $\ ext{cos } \theta$:

cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta

cos⁡θ=±1−sin⁡2θ\cos \theta = \pm\sqrt{1 - \sin^2 \theta}

Choosing the Correct Sign: Quadrants to the Rescue Again!

Notice the $\pm$ sign in front of the square root. This means there are two possible values for $\ ext{sin } \theta$ or $\ ext{cos } \theta$, one positive and one negative. This is where our knowledge of quadrants comes in handy again. By knowing the quadrant in which $\ heta$ lies, we can determine the correct sign for the trigonometric function.

For instance, if we know $\ ext{cos } \theta$ is negative and $\ heta$ is in the second quadrant, we know that $\ ext{sin } \theta$ must be positive. So, we would choose the positive square root when solving for $\ ext{sin } \theta$.

Beyond the Basics: Other Forms of the Pythagorean Identity

The Pythagorean identity also has some useful variations that can be derived by dividing both sides of the original identity by $\cos^2 \theta$ or $\sin^2 \theta$. These variations involve the other trigonometric functions, tangent, secant, and cosecant.

Dividing by $\cos^2 \theta$

If we divide both sides of $\sin^2 \theta + \cos^2 \theta = 1$ by $\cos^2 \theta$, we get:

sin⁡2θcos⁡2θ+cos⁡2θcos⁡2θ=1cos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}

Since $\frac{\sin \theta}{\cos \theta} = \tan \theta$ and $\frac{1}{\cos \theta} = \sec \theta$, this simplifies to:

tan⁡2θ+1=sec⁡2θ\tan^2 \theta + 1 = \sec^2 \theta

Dividing by $\sin^2 \theta$

Similarly, if we divide both sides of $\sin^2 \theta + \cos^2 \theta = 1$ by $\sin^2 \theta$, we get:

sin⁡2θsin⁡2θ+cos⁡2θsin⁡2θ=1sin⁡2θ\frac{\sin^2 \theta}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}

Since $\frac{\cos \theta}{\sin \theta} = \cot \theta$ and $\frac{1}{\sin \theta} = \csc \theta$, this simplifies to:

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

These variations of the Pythagorean identity can be incredibly useful in solving more complex trigonometric problems.

Practice Makes Perfect: Tackling Similar Problems

Now that we've walked through the solution and explored the underlying concepts, the best way to solidify your understanding is to practice! Let's tackle a few similar problems to sharpen your skills.

Problem 1

If $\sin \theta = \frac{3}{5}$ and $\cos \theta < 0$, find the value of $\tan \theta$.

Solution

  1. Identify the Quadrant: Sine is positive, and cosine is negative, which means $\ heta$ is in the second quadrant.
  2. Use the Pythagorean Identity: We can use $\sin^2 \theta + \cos^2 \theta = 1$ to find $\cos \theta$:

    (35)2+cos⁡2θ=1\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1

    925+cos⁡2θ=1\frac{9}{25} + \cos^2 \theta = 1

    cos⁡2θ=1−925=1625\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}

    cos⁡θ=±1625=±45\cos \theta = \pm\sqrt{\frac{16}{25}} = \pm\frac{4}{5}

    Since cosine is negative in the second quadrant, $\ ext{cos } \theta = -\frac{4}{5}$.
  3. Find Tangent: Use the definition of tangent, $\tan \theta = \frac{\sin \theta}{\cos \theta}$:

    tan⁡θ=35−45=−34\tan \theta = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}

So, $\tan \theta = -\frac{3}{4}$.

Problem 2

If $\tan \theta = 2$ and $\cos \theta < 0$, find the value of $\ ext{sin } \theta$.

Solution

  1. Identify the Quadrant: Tangent is positive, and cosine is negative, which means $\ heta$ is in the third quadrant.
  2. Use a Pythagorean Identity Variation: We can use $\tan^2 \theta + 1 = \sec^2 \theta$ to find $\sec \theta$:

    22+1=sec⁡2θ2^2 + 1 = \sec^2 \theta

    5=sec⁡2θ5 = \sec^2 \theta

    sec⁡θ=±5\sec \theta = \pm\sqrt{5}

    Since cosine is negative in the third quadrant, secant (which is 1/cosine) is also negative, so $\ ext{sec } \theta = -\sqrt{5}$.
  3. Find Cosine: Use the definition of secant, $\ ext{sec } \theta = \frac{1}{\cos \theta}$:

    cos⁡θ=1sec⁡θ=1−5=−15\cos \theta = \frac{1}{\sec \theta} = \frac{1}{-\sqrt{5}} = -\frac{1}{\sqrt{5}}

  4. Find Sine: Use the definition of tangent, $\tan \theta = \frac{\sin \theta}{\cos \theta}$:

    2=sin⁡θ−152 = \frac{\sin \theta}{-\frac{1}{\sqrt{5}}}

    sin⁡θ=2⋅(−15)=−25\sin \theta = 2 \cdot \left(-\frac{1}{\sqrt{5}}\right) = -\frac{2}{\sqrt{5}}

So, $\sin \theta = -\frac{2}{\sqrt{5}}$.

Key Takeaways for Solving These Problems

  • Identify the Quadrant First: This helps you determine the signs of the trigonometric functions.
  • Use the Pythagorean Identity (or its variations): This allows you to relate sine, cosine, and other trigonometric functions.
  • Choose the Correct Sign: Don't forget to consider the quadrant when taking square roots.
  • Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become with these concepts.

Conclusion: Mastering Trigonometric Puzzles

And there you have it! We've successfully navigated the world of trigonometric functions, quadrants, and the Pythagorean identity. By understanding these concepts, you can confidently tackle problems where you need to find the value of one trigonometric function given information about others. It's all about piecing together the clues and using the right tools.

Remember, trigonometry might seem daunting at first, but with a little practice and a solid understanding of the fundamentals, you'll be solving these puzzles like a pro. Keep exploring, keep practicing, and you'll be amazed at what you can achieve! You've got this!