Question

In Exercises 11-16, evaluate the expression given that \(\cos a=\frac{4}{5}\) with \(0

Short answer

-36/85

Step-by-step solution

Recall the appropriate trigonometric identity

Recall the sine addition formula, which states: \(\sin(a+b) = \sin a \cos b + \cos a \sin b\)

Compute the missing sine and cosine

Use the Pythagorean identity to compute the missing sine and cosine. The Pythagorean identity is: \(\sin^2 \theta + \cos^2 \theta = 1\) Therefore, \(\sin a = \sqrt{1 - \cos^2 a} = \sqrt{1 - (\frac{4}{5})^2} = \frac{3}{5}\), and \(\cos b = \sqrt{1 - \sin^2 b} = \sqrt{1 - (\frac{-15}{17})^2} = \frac{8}{17}\). Note: The negative square root for \(\sin a\) and \(\cos b\) should be discarded, because a is in the first quadrant (where sin is positive) and b is in the fourth quadrant (where cosine is positive).

Substitute values into the sum of angles formula

Substitute \(\sin a\), \(\cos a\), \(\sin b\), and \(\cos b\) into the sine sum of angles formula: \(\sin(a+b) = \sin a \cos b + \cos a \sin b = \frac{3}{5} \cdot \frac{8}{17} + \frac{4}{5} \cdot \frac{-15}{17} = \frac{24}{85} - \frac{60}{85} = -\frac{36}{85}\)

Simplify the expression

The above result can be simplified or left as it is. In this case, it is already in the simplest form.

Key concepts

Sine Addition Formula

The sine addition formula is a powerful tool used in trigonometry to determine the sine of the sum of two angles. This formula is expressed mathematically as \( \sin(a+b) = \sin a \cos b + \cos a \sin b \). Understanding how to apply this formula is essential when dealing with angles in different quadrants.

• \( \sin a \) and \( \sin b \) represent the sine values of angles \( a \) and \( b \) respectively.
• \( \cos a \) and \( \cos b \) represent the cosine values of these angles.

When using the sine addition formula, you must first find the sine and cosine of both individual angles. Then, substitute them into the formula to calculate the sine of their sum. This approach is particularly useful when you know the values of sine and cosine for given angles but need to find the sine of their sum.
In the problem above, using the sine addition formula helped simplify the calculation needed to find \( \sin(a+b) \). By substituting known values, the expression was evaluated quickly and accurately.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship between the sine and cosine of an angle, expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). This equation is a crucial part of trigonometry because it allows for the calculation of one trigonometric function if the other is known.

• When \( \cos a = \frac{4}{5} \), the Pythagorean identity helps find \( \sin a \).
• Similarly, it helps determine \( \cos b \) when \( \sin b = -\frac{15}{17} \).

To implement this identity, just rearrange it based on the trigonometric function available. For instance, if \( \cos a \) is known, \( \sin a \) can be found using \( \sin a = \sqrt{1 - \cos^2 a} \). The positive or negative outcome of the square root is determined by the angle's quadrant.
This identity is not only key for solving step-by-step problems but is also a building block for many more advanced trigonometric identities and formulas.

Sum of Angles Formula

The sum of angles formula encompasses several trigonometric identities that reveal the relationship between angles. One of these is the sine addition formula, \( \sin(a+b) \) as discussed earlier.
Another is the cosine sum formula, \( \cos(a+b) = \cos a \cos b - \sin a \sin b \), which provides a way to calculate the cosine of the sum of two angles using their individual sine and cosine.
The sum of angles formulas are very helpful in various applications:

• They allow the conversion of complex angle expressions into simpler forms.
• They enable calculations in many geometric and trigonometric problems involving angle measures.

In the context of this exercise, the sine sum of angles formula was effectively used to evaluate \( \sin(a+b) \), making the problem-solving process faster and more efficient.

Trigonometric Functions

Trigonometric functions play a central role in trigonometry by expressing the relationships between the angles and sides of triangles. The basic trigonometric functions include sine, cosine, and tangent, each defined based on ratios of triangle sides:

• Sine (\( \sin \)): Opposite side over hypotenuse.
• Cosine (\( \cos \)): Adjacent side over hypotenuse.
• Tangent (\( \tan \)): Opposite side over adjacent side.

These functions are versatile, allowing you to explore properties of angles both within triangles (using known sides) and on the unit circle (with coordinates).

The Unit Circle

Understanding trigonometric functions involves the unit circle, where the radius is 1. It helps visualize how these functions behave over different angles.
In unit circle terms:

• \( \sin \theta \) is the y-coordinate.
• \( \cos \theta \) is the x-coordinate.

Application in Problem Solving

In the given problem, identifying \( \sin a \) and \( \cos b \) was crucial for using the sine addition formula. Grasping these foundational functions is key to tackling a variety of trigonometric problems.