Problem 56 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sin w}{\sin w+\cos w}=\frac{\tan w}{1+\tan w} $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \frac{\sin w}{\sin w + \cos w} \).

Step 1: Write Down the Given Identity

We have to prove that \( \frac{\sin w}{\sin w + \cos w} = \frac{\tan w}{1 + \tan w} \). We will work on the left side and try to transform it to look like the right side.

Step 2: Express Left Side in Terms of Sine and Cosine

Notice that \( \tan w = \frac{\sin w}{\cos w} \). To transform \( \frac{\tan w}{1 + \tan w} \), first express \( \tan w \) in terms of sine and cosine: \( \tan w = \frac{\sin w}{\cos w} \). Therefore, \( 1 + \tan w = 1 + \frac{\sin w}{\cos w} \).

Step 3: Simplify the Denominator on Right Side

The expression for \( 1 + \tan w \) becomes \( \frac{\cos w + \sin w}{\cos w} \). Substitute this back into the right side to obtain: \[ \frac{\frac{\sin w}{\cos w}}{\frac{\cos w + \sin w}{\cos w}} \].

Step 4: Simplify the Right Side

Cancel the common terms in the numerator and denominator to get \[ \frac{\sin w}{\cos w + \sin w} \].

Step 5: Compare Both Sides

Now the right side is \( \frac{\sin w}{\cos w + \sin w} \). Notice that this is exactly the same as the left side \( \frac{\sin w}{\sin w + \cos w} \), showing they are equal.

Key Concepts

The Sine Function

The sine function is an essential building block in trigonometry. It is commonly denoted as \( \sin \theta \), where \( \theta \) is the angle in a right-angled triangle. In such a triangle, the sine of an angle is the ratio of the length of the side opposite to the angle to the hypotenuse.
\[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \]
Understanding this ratio is crucial when working with trigonometric identities, which often involve transforming expressions to show equivalence. In our exercise, \( \sin w \) represents the sine of angle \( w \), playing a pivotal role in rewriting and simplifying expressions to verify the given identity.
Trigonometry extends beyond triangles and angles. It's used for modeling periodic phenomena like sound waves and tides. Sine, alongside cosine and tangent, helps translate these natural patterns into mathematical terms.

The Cosine Function

Cosine is another fundamental trigonometric function, closely related to the sine function. It is denoted as \( \cos \theta \). In a right-angled triangle, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
\[ \cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} \]
Much like the sine function, cosine appears frequently in trigonometric identities and transformations. It complements the sine function by addressing the other non-hypotenuse side of the triangle. This relationship is essential in our exercise, where cosine assists in expressing the tangent function in terms of sine and cosine.

• Cosine complements sine in circular functions, with \( \cos \theta = \sin (90^\circ - \theta) \).
• It's vital for understanding the full range of trigonometric applications, from physics to engineering.

Understanding cosine is as crucial as understanding sine in the realm of trigonometric identities and their verification.

The Tangent Function

The tangent function is represented by \( \tan \theta \) and is perhaps most notable for its relationship to both sine and cosine. In a right-angled triangle, the tangent of an angle is the ratio of the sine of the angle to the cosine of the angle.
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
This ratio is significant when validating identities such as the one in our exercise, where expressing \( \tan w \) in terms of \( \sin w \) and \( \cos w \) is a necessary step in proving equality.
The tangent function naturally arises in slopes and angles, providing insights into their behaviors. Often used in physics and engineering, tangent simplifies angular analysis in various contexts.

• It helps in converting between angles and their respective trigonometric values.
• Understanding the interplay between tangent, sine, and cosine is vital for grasping the full picture of trigonometric identities.

Mastering tangent aids significantly in handling complex trigonometric transformations.