Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the trigonometric expression when \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} \cdot(\text { Both } u\) and \(v\) are in Quadrant II.) $$\sin (u+v)$$

Step-by-Step Solution

Answer

The exact value of \( \sin (u+v) \) is -\frac{56}{65}.

Step 1: Determine \(\cos u\) and \(\sin v\)

Using the Pythagorean identity \( \sin^2 u + \cos^2 u = 1 \) and \( \sin^2 v + \cos^2 v = 1 \), we can find the unknown sine and cosine by solving for cos(u) and sin(v). Given that in quadrant II, cosine is negative and sine is positive, we get \(\cos u = -\sqrt{1- \sin^2 u} = -\sqrt{1- \left(\frac{5}{13}\right)^2} = -\frac{12}{13}\) and \(\sin v = \sqrt{1- \cos^2 v} = \sqrt{1- \left(-\frac{3}{5}\right)^2} = \frac{4}{5}\).

Step 2: Use the sine addition formula

We substitute the values of \( \sin u \), \( \cos u \), \( \sin v \), and \( \cos v \) into the formula for sine of addition of two angles \( \sin (u+v) = \sin u \cos v + \cos u \sin v \). This gives us \( \sin (u+v) = \frac{5}{13} \cdot -\frac{3}{5} + -\frac{12}{13} \cdot \frac{4}{5} = -\frac{56}{65} \).

Step 3: Final answer

The exact value of \( \sin (u+v) \) is found to be -\frac{56}{65}.

Key Concepts

Sine Addition Formula

The sine addition formula is a crucial tool in trigonometry. It helps us find the sine of the sum of two angles. The formula is given by: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] This equation is derived from the unit circle and the definitions of sine and cosine. In our exercise, this formula allows us to find \( \sin(u+v) \), given the sine and cosine values for angles \(u\) and \(v\). Let's break this down:

• For angle \(u\), we know \(\sin u\) and found \(\cos u\) using the Pythagorean identity.
• For angle \(v\), we know \(\cos v\) and found \(\sin v\) similarly.
• Substituting these into the sine addition formula, we calculate \(\sin(u+v)\).

By understanding this formula, you can easily solve complex trigonometric problems involving angle sums in various quadrants.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It states: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This identity is derived from the Pythagorean theorem, applying to the relations of the sides of a right triangle inscribed in the unit circle. In the context of our problem:

• We used the identity to find unknown trigonometric values of \(\cos u\) and \(\sin v\).
• We know \(\sin u = \frac{5}{13}\), and since \(u\) is in the second quadrant where cosine values are negative, \(\cos u = -\sqrt{1-\left(\frac{5}{13}\right)^2} = -\frac{12}{13}\).
• Similarly, with \(\cos v = -\frac{3}{5}\), and \(v\) also in the second quadrant, \(\sin v = \sqrt{1-\left(-\frac{3}{5}\right)^2} = \frac{4}{5}\).

By applying the Pythagorean identity, you gain a complete set of trigonometric functions for an angle, which is especially useful in problems involving trigonometric identities and transformations.

Quadrant II Trigonometric Functions

Understanding trigonometric functions in Quadrant II is key to solving problems accurately. Quadrant II is characterized by angles between 90° and 180°. In this quadrant:

• Sine functions are positive.
• Cosine functions are negative.
• Tangent functions are negative.

This knowledge helps us correctly determine the signs of the trigonometric values. For our exercise:

• We knew \(\sin u\) is positive and \(\cos u\) negative, given \(u\) is in Quadrant II.
• Similarly, \(\cos v\) is negative and \(\sin v\) positive.

Recognizing the sign of these functions according to the quadrant ensures the accuracy when applying trigonometric formulas, as seen in calculating \(\sin(u+v)\). Remembering the basics about quadrants can greatly simplify complex trigonometric calculations.