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If Sin() = 8/17 Where 0 < < /2 And Cos() = 5/13 Where 3/2 < < 2, Find The Exact Values Of

If Sin() = 8/17 Where 0 < < π/2 And Cos() = 5/13 Where 3π/2 < < 2π, Find The Exact Values Of

Understanding the exact values of sine and cosine functions for given angles is fundamental in trigonometry. These values are crucial for solving various mathematical problems, including those involving right triangles, unit circles, and trigonometric identities. In this article, we will explore a detailed approach to find the exact values of trigonometric functions based on the given sine and cosine ratios, considering the specific quadrants indicated by the angle ranges.

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Understanding the Given Data and Constraints

Before diving into calculations, it's essential to understand the given information:


  • Sin() = 8/17 with 0 < < π/2

  • Cos() = 5/13 with 3π/2 < < 2π


These indicate two separate angles, each located in different quadrants, with their respective sine and cosine values.

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Analyzing the First Angle: Sin() = 8/17, 0 < < π/2

Quadrant and Significance

  • The angle is in the first quadrant because 0 < < π/2 (0 to 90 degrees).
  • In the first quadrant, both sine and cosine are positive.

Determining the Exact Values

  • Given sin θ = 8/17, which means the ratio of the opposite side to the hypotenuse in a right triangle is 8/17.
  • To find the cosine and tangent, we need to determine the adjacent side and other trigonometric ratios.

Using the Pythagorean Theorem

  • The hypotenuse, h = 17.
  • Opposite side, opposite = 8.
  • Adjacent side, adjacent = ?
Applying the Pythagorean theorem: \[ \text{adjacent}^2 + \text{opposite}^2 = \text{hypotenuse}^2 \] \[ \text{adjacent}^2 + 8^2 = 17^2 \] \[ \text{adjacent}^2 + 64 = 289 \] \[ \text{adjacent}^2 = 289 - 64 = 225 \] \[ \text{adjacent} = \sqrt{225} = 15 \]

Since the angle is in the first quadrant, adjacent is positive:
\[
\text{adjacent} = 15
\]

Calculating Cosine and Other Ratios

  • cos θ = adjacent / hypotenuse = 15/17
  • tan θ = opposite / adjacent = 8/15
Summary for the first angle:
  • \(\sin \theta = \frac{8}{17}\)
  • \(\cos \theta = \frac{15}{17}\)
  • \(\tan \theta = \frac{8}{15}\)
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Analyzing the Second Angle: Cos() = 5/13, 3π/2 < < 2π

Quadrant and Significance

  • The angle lies in the fourth quadrant, since 3π/2 < < 2π (270° to 360°).
  • In the fourth quadrant:
  • cosine is positive
  • sine is negative

Determining the Exact Values

  • Given cos φ = 5/13, which represents the adjacent over hypotenuse ratio.

Applying the Pythagorean Theorem

  • Hypotenuse, h = 13
  • Adjacent side, adjacent = 5
  • Opposite side, opposite = ?
Calculating the opposite side: \[ \text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2 \] \[ \text{opposite}^2 + 5^2 = 13^2 \] \[ \text{opposite}^2 + 25 = 169 \] \[ \text{opposite}^2 = 169 - 25 = 144 \] \[ \text{opposite} = \pm 12 \]

Since the angle is in the fourth quadrant, where sine is negative:
\[
\sin \phi = -\frac{12}{13}
\]

Summary for the second angle:


  • \(\cos \phi = \frac{5}{13}\)

  • \(\sin \phi = -\frac{12}{13}\)


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Summary of Known Values

| Function | First Angle | Second Angle | |----------|-------------------------|--------------------------| | \(\sin\) | \(\frac{8}{17}\) | \(-\frac{12}{13}\) | | \(\cos\) | \(\frac{15}{17}\) | \(\frac{5}{13}\) | | \(\tan\) | \(\frac{8}{15}\) | \(-\frac{12}{5}\) |

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Finding Other Trigonometric Ratios

Once the sine and cosine values are known, you can compute other ratios such as tangent, cotangent, secant, and cosecant directly.

Calculating Tangent and Cotangent

  • For the first angle:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{8/17}{15/17} = \frac{8}{15} \]
  • For the second angle:
\[ \tan \phi = \frac{\sin \phi}{\cos \phi} = \frac{-12/13}{5/13} = -\frac{12}{5} \]

Calculating Secant and Cosecant

  • Secant is reciprocal of cosine:
  • First angle:
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{17}{15} \]
  • Second angle:
\[ \sec \phi = \frac{1}{\cos \phi} = \frac{13}{5} \]
  • Cosecant is reciprocal of sine:
  • First angle:
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{17}{8} \]
  • Second angle:
\[ \csc \phi = \frac{1}{\sin \phi} = -\frac{13}{12} \]

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Applications of These Exact Values in Trigonometry

Knowing the exact values of sine, cosine, and tangent allows for solving complex problems involving:


  • Trigonometric identities: verifying identities like Pythagorean, angle sum/difference, double angle, etc.

  • Solving triangles: determining unknown sides or angles in right and oblique triangles.

  • Graphing trigonometric functions: understanding their behavior based on known values.

  • Calculus applications: derivatives and integrals involving trigonometric functions.


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Verifying the Results with Trigonometric Identities

Let's verify some identities using the calculated values:


  • Pythagorean Identity:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]
For the first angle:
\[
\left(\frac{8}{17}\right)^2 + \left(\frac{15}{17}\right)^2 = \frac{64}{289} + \frac{225}{289} = \frac{289}{289} = 1
\]

  • For the second angle:

\[
\left(-\frac{12}{13}\right)^2 + \left(\frac{5}{13}\right)^2 = \frac{144}{169} + \frac{25}{169} = \frac{169}{169} = 1
\]

This confirms the accuracy of the calculated values.

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Conclusion

In this comprehensive guide, we have explored how to determine the exact values of trigonometric functions given specific sine and cosine ratios for angles located in different quadrants. By applying the Pythagorean theorem, understanding quadrant signs, and calculating reciprocal functions, we can accurately find all primary and secondary trigonometric ratios.

Summary:


  • For the angle in the first quadrant with \(\sin \theta = 8/17\), the exact values are:

\[
\sin \theta = \frac{8}{17}, \quad \cos \theta = \frac{15}{17}, \quad \tan \theta = \frac{8}{15}
\]

  • For the angle in the fourth quadrant with

Frequently Asked Questions

Given sin(θ) = 8/17 where 0 < θ < π/2, and cos(φ) = 5/13 where 3π/2 < φ < 2π, find the exact value of sin(θ + φ).
Using the sine addition formula: sin(θ + φ) = sin θ cos φ + cos θ sin φ.

First, find cos θ: cos θ = √(1 - sin² θ) = √(1 - (8/17)²) = √(1 - 64/289) = √(225/289) = 15/17.

Next, find sin φ: sin φ = √(1 - cos² φ) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13.

Since φ is in the 3rd or 4th quadrant (3π/2 < φ < 2π), sin φ is negative: sin φ = -12/13.

Similarly, cos φ = 5/13 (positive in 4th quadrant).

Now, sin(θ + φ) = (8/17)(5/13) + (15/17)(-12/13) = (40/221) - (180/221) = -140/221.

Therefore, sin(θ + φ) = -140/221.
If sin θ = 8/17 with 0 < θ < π/2, what is the exact value of cos θ?
cos θ = √(1 - sin² θ) = √(1 - (8/17)²) = √(1 - 64/289) = √(225/289) = 15/17.
Given cos φ = 5/13 where 3π/2 < φ < 2π, find the exact value of tan φ.
tan φ = sin φ / cos φ.

Find sin φ: sin φ = -12/13 (since φ is in the 4th quadrant where sine is negative).

Thus, tan φ = (-12/13) / (5/13) = -12/5.
With the given sin θ = 8/17, what is the exact value of cos(2θ)?
Using the double angle formula: cos 2θ = 1 - 2 sin² θ.

sin² θ = (8/17)² = 64/289.

Therefore, cos 2θ = 1 - 2(64/289) = 1 - 128/289 = (289/289) - (128/289) = 161/289.
Using the given cos φ = 5/13 where 3π/2 < φ < 2π, find the exact value of sin(π/2 + φ).
sin(π/2 + φ) = cos φ.

Since cos φ = 5/13, sin(π/2 + φ) = 5/13.
Given sin θ = 8/17 with 0 < θ < π/2, and cos φ = 5/13 where 3π/2 < φ < 2π, find the exact value of sin(θ - φ).
Using the sine difference formula: sin(θ - φ) = sin θ cos φ - cos θ sin φ.

We already have sin θ = 8/17, cos θ = 15/17.

For φ, sin φ = -12/13, cos φ = 5/13.

Therefore, sin(θ - φ) = (8/17)(5/13) - (15/17)(-12/13) = (40/221) + (180/221) = 220/221.

Hence, sin(θ - φ) = 220/221.

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