Finding the Value of tan θ Given sin θ

The question asks us to find the value of tan θ given that sin θ is \( \frac{8}{17} \). We can solve this problem using trigonometric identities or by constructing a right-angled triangle.

Understanding Trigonometric Ratios

In a right-angled triangle, the primary trigonometric ratios are defined as:

• sin θ = \( \frac{\text{Opposite side}}{\text{Hypotenuse}} \)
• cos θ = \( \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)
• tan θ = \( \frac{\text{Opposite side}}{\text{Adjacent side}} \)

Also, there is a fundamental identity relating these ratios: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Method 1: Using Trigonometric Identities

We are given \( \sin \theta = \frac{8}{17} \). We can use the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \), to find the value of cos θ.

Substitute the value of sin θ into the identity:

\( \left(\frac{8}{17}\right)^2 + \cos^2 \theta = 1 \)

\( \frac{64}{289} + \cos^2 \theta = 1 \)

Now, isolate \( \cos^2 \theta \):

\( \cos^2 \theta = 1 - \frac{64}{289} \)

To subtract, find a common denominator:

\( \cos^2 \theta = \frac{289}{289} - \frac{64}{289} \)

\( \cos^2 \theta = \frac{289 - 64}{289} \)

\( \cos^2 \theta = \frac{225}{289} \)

Take the square root of both sides to find cos θ. Assuming θ is in the first quadrant where cosine is positive:

\( \cos \theta = \sqrt{\frac{225}{289}} \)

\( \cos \theta = \frac{\sqrt{225}}{\sqrt{289}} \)

\( \cos \theta = \frac{15}{17} \)

Now that we have sin θ and cos θ, we can find tan θ using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \):

\( \tan \theta = \frac{\frac{8}{17}}{\frac{15}{17}} \)

\( \tan \theta = \frac{8}{17} \times \frac{17}{15} \)

Cancel out the 17s:

\( \tan \theta = \frac{8}{15} \)

Method 2: Using a Right-Angled Triangle

We know that \( \sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} \). Given \( \sin \theta = \frac{8}{17} \), we can consider a right-angled triangle where the length of the side opposite to angle θ is 8 units and the length of the hypotenuse is 17 units.

Let the opposite side be \( o \), the adjacent side be \( a \), and the hypotenuse be \( h \).

• \( o = 8 \)
• \( h = 17 \)

We need to find the length of the adjacent side \( a \). We can use the Pythagorean theorem, which states that \( o^2 + a^2 = h^2 \) in a right-angled triangle.

Substitute the known values:

\( 8^2 + a^2 = 17^2 \)

\( 64 + a^2 = 289 \)

Now, solve for \( a^2 \):

\( a^2 = 289 - 64 \)

\( a^2 = 225 \)

Take the square root of both sides to find \( a \):

\( a = \sqrt{225} \)

\( a = 15 \)

So, the length of the adjacent side is 15 units.

Now we have the lengths of the opposite and adjacent sides:

Finally, we can find tan θ using its definition:

\( \tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{o}{a} \)

\( \tan \theta = \frac{8}{15} \)

Conclusion: Value of tan θ

Both methods give the same result. If \( \sin \theta = \frac{8}{17} \), then the value of \( \tan \theta \) is \( \frac{8}{15} \).

Revision Table: Basic Trigonometric Ratios

Additional Information: The Pythagorean Identity

The identity \( \sin^2 \theta + \cos^2 \theta = 1 \) is one of the most fundamental identities in trigonometry. It comes directly from the Pythagorean theorem.

Consider a right-angled triangle with angle θ. Let the opposite side be \( o \), the adjacent side be \( a \), and the hypotenuse be \( h \).

According to the Pythagorean theorem: \( o^2 + a^2 = h^2 \).

Divide the entire equation by \( h^2 \):

\( \frac{o^2}{h^2} + \frac{a^2}{h^2} = \frac{h^2}{h^2} \)

\( \left(\frac{o}{h}\right)^2 + \left(\frac{a}{h}\right)^2 = 1 \)

Recall the definitions: \( \sin \theta = \frac{o}{h} \) and \( \cos \theta = \frac{a}{h} \).

Substituting these definitions gives:

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

Which is commonly written as \( \sin^2 \theta + \cos^2 \theta = 1 \).

This identity is useful for finding the value of sin θ if cos θ is known, and vice versa, and is a key tool in solving many trigonometric problems.