Solving Trigonometric Functions Given Tangent Theta Equals 15 8

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Hey guys! Let's dive into a fun trigonometry problem where we're given that $\tan \theta = \frac{15}{8}$ and we need to figure out which of the following statements are true:

  • A. $\csc \theta = \frac{17}{15}$
  • B. $\cot \theta = \frac{8}{15}$
  • C. $\cos \theta = \frac{15}{17}$
  • D. $\sec \theta = \frac{17}{8}$

This is a classic trigonometry question that tests our understanding of the relationships between trigonometric functions. We'll use the given information and some fundamental trigonometric identities to solve this. So, grab your thinking caps, and let's get started!

Understanding the Basics: SOH CAH TOA and Trigonometric Identities

Before we jump into solving the problem, it's super important to refresh our memory on some basic trigonometric concepts. Remember the SOH CAH TOA mnemonic? This handy tool helps us recall the definitions of sine, cosine, and tangent in a right-angled triangle:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Since we're given $\tan \theta = \frac{15}{8}$, we know that in a right-angled triangle, the side opposite to angle θ is 15 units, and the side adjacent to angle θ is 8 units. This is our starting point, guys. We can visualize this as a right triangle where the vertical side is 15 and the horizontal side is 8. But wait, we need the hypotenuse to figure out sine, cosine, cosecant, and secant! That's where the Pythagorean theorem comes to our rescue.

The Pythagorean Theorem to the Rescue

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, this is expressed as:

a2+b2=c2a^2 + b^2 = c^2

where a and b are the lengths of the two shorter sides (legs) of the triangle, and c is the length of the hypotenuse. In our case, a = 8 and b = 15. Let's plug these values into the Pythagorean theorem to find the hypotenuse c:

82+152=c28^2 + 15^2 = c^2

64+225=c264 + 225 = c^2

289=c2289 = c^2

Taking the square root of both sides, we get:

c=289=17c = \sqrt{289} = 17

So, the hypotenuse of our right-angled triangle is 17 units. Now we have all three sides of the triangle: opposite = 15, adjacent = 8, and hypotenuse = 17. We're one step closer, guys!

Beyond SOH CAH TOA: Cosecant, Cotangent, and Secant

Now that we know the three sides of our triangle, let's talk about cosecant, cotangent, and secant. These are the reciprocal trigonometric functions:

  • Cosecant (csc) is the reciprocal of sine: $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$
  • Cotangent (cot) is the reciprocal of tangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}}$
  • Secant (sec) is the reciprocal of cosine: $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$

These reciprocal relationships are crucial for solving our problem. We already know the opposite, adjacent, and hypotenuse, so we can easily calculate these reciprocal functions. Let's do it!

Solving the Problem: Finding Cosecant, Cotangent, Cosine, and Secant

Okay, guys, now comes the fun part – actually solving for the trigonometric values! We'll use the definitions and the side lengths we found to determine which of the given statements are true.

Evaluating Option A: Is csc θ = 17/15?

Let's calculate the cosecant of θ using our definition:

csc⁡θ=HypotenuseOpposite=1715\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{17}{15}

So, option A, $\csc \theta = \frac{17}{15}$, is TRUE! We're off to a good start, guys!

Evaluating Option B: Is cot θ = 8/15?

Now let's find the cotangent of θ:

cot⁡θ=AdjacentOpposite=815\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{8}{15}

Option B, $\cot \theta = \frac{8}{15}$, is also TRUE! We're on a roll here!

Evaluating Option C: Is cos θ = 15/17?

Time to calculate the cosine of θ:

cos⁡θ=AdjacentHypotenuse=817\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{17}

Wait a minute! Option C states that $\cos \theta = \frac{15}{17}$, but we calculated $\cos \theta = \frac{8}{17}$. So, option C is FALSE. Good catch!

Evaluating Option D: Is sec θ = 17/8?

Finally, let's calculate the secant of θ:

sec⁡θ=HypotenuseAdjacent=178\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{17}{8}

Option D, $\sec \theta = \frac{17}{8}$, is TRUE! We've nailed it, guys!

The Final Verdict: Which Statements Apply?

After carefully calculating each trigonometric value, we've determined that the following statements are true:

  • A. $\csc \theta = \frac{17}{15}$
  • B. $\cot \theta = \frac{8}{15}$
  • D. $\sec \theta = \frac{17}{8}$

Option C, $\cos \theta = \frac{15}{17}$, is false. We successfully navigated this trigonometric problem by using SOH CAH TOA, the Pythagorean theorem, and the definitions of reciprocal trigonometric functions. Great job, guys! This kind of problem really reinforces our understanding of these core concepts.

Key Takeaways and Practice Tips

Alright, guys, let's wrap things up with some key takeaways and tips for practicing these types of trigonometry problems:

  • Master SOH CAH TOA: This mnemonic is your best friend for remembering the definitions of sine, cosine, and tangent.
  • Know Your Reciprocals: Understand the relationships between sine and cosecant, cosine and secant, and tangent and cotangent. This is crucial for efficiently solving problems.
  • Pythagorean Theorem is Key: You'll often need to use the Pythagorean theorem to find the missing side of a right-angled triangle.
  • Visualize the Triangle: Drawing a right-angled triangle and labeling the sides can make the problem much clearer.
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with these concepts. Try solving different problems with varying given information.

Trigonometry can seem intimidating at first, but with a solid understanding of the basics and consistent practice, you'll be solving these problems like a pro in no time! Keep up the great work, and remember, the world of trigonometry is full of fascinating relationships and applications. So, keep exploring and keep learning!

Further Exploration: Trigonometric Identities and Applications

For those of you who are eager to delve deeper into trigonometry, there's a whole world of trigonometric identities and applications to explore! Trigonometric identities are equations that are true for all values of the variables involved. They are incredibly useful for simplifying expressions, solving equations, and proving other results in trigonometry and calculus.

Some fundamental trigonometric identities include:

  • Pythagorean Identities: These are derived from the Pythagorean theorem and are among the most important identities:

    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

    • 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

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

  • Reciprocal Identities: We've already touched on these, but it's worth reiterating:

    • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

    • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

    • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

  • Quotient Identities:

    • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

    • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}

Understanding and being able to apply these identities is a game-changer when tackling more complex trigonometric problems. They allow you to manipulate expressions, simplify equations, and ultimately find solutions more efficiently.

Real-World Applications of Trigonometry

But trigonometry isn't just about abstract equations and identities; it has a plethora of real-world applications! From navigation and surveying to engineering and physics, trigonometry plays a vital role in many fields. Here are just a few examples:

  • Navigation: Trigonometry is used extensively in navigation to determine distances, directions, and positions. Sailors, pilots, and even GPS systems rely on trigonometric principles.
  • Surveying: Surveyors use trigonometry to measure land, create maps, and determine property boundaries.
  • Engineering: Engineers use trigonometric functions to design structures, calculate forces, and analyze stresses.
  • Physics: Trigonometry is fundamental in physics for analyzing wave motion, projectile motion, and other phenomena.
  • Astronomy: Astronomers use trigonometry to measure distances to stars and planets and to understand celestial mechanics.

So, as you can see, the knowledge of trigonometry you're gaining is not just for passing exams; it's a valuable tool that can be applied in many exciting and practical ways. Keep exploring, keep questioning, and keep applying what you've learned. The world is your trigonometric oyster!