Question

Evaluate the expression given that \(\cos a=\frac{4}{5}\) with \(0

Short answer

\(\cos (a-b)= -\frac{77}{85}\)

Step-by-step solution

Determine the values of sin a and cos b

Given that \(\cos a = \frac{4}{5}\) and \(\sin b = -\frac{15}{17}\), calculate \(\sin a\) and \(\cos b\) using the relationship: \(\sin^2 \theta + \cos^2 \theta = 1\), which implies \(\sin \theta = \sqrt{1 - \cos^2 \theta}\) and \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). Now based on the given angles range, take the appropriate sign for the roots, yielding \(\sin a = \frac{3}{5}\) and \(\cos b = -\frac{8}{17}\).

Evaluate the expression

Substitute the values of \(\cos a\), \(\cos b\), \(\sin a\) and \(\sin b\) into the formula \(\cos (a-b) = \cos a \cos b + \sin a \sin b\). This leads to \(\cos (a-b) = \frac{4}{5} \times -\frac{8}{17} + \frac{3}{5} \times -\frac{15}{17} = -\frac{32}{85} - \frac{45}{85} = -\frac{77}{85}\).

Key concepts

Cosine of Angle Difference

Evaluating the cosine of an angle difference involves one of the key formulas in trigonometry: \( \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \). This formula is derived from the trigonometric sum and difference identities and is essential for solving problems where two angles are involved, such as \(\cos (a-b)\) in the given exercise.

To use this identity effectively, one must remember to retain the signs of each trigonometric function based on the quadrants in which the angles \(\alpha\) and \(\beta\) lie. As we know from the ranges given for \(a\) and \(b\) in the original problem, \(a\) is in the first quadrant where both sine and cosine are positive, while \(b\) is in the fourth quadrant where sine is negative, and cosine is positive. This understanding helps in ascertaining the correct signs when substituting the values into the identity.

Pythagorean Trigonometric Identity

The Pythagorean trigonometric identity is a fundamental concept in trigonometry and states that for any angle \(\theta\), the sum of the squares of the sine and cosine of \(\theta\) equals one: \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is derived from the Pythagorean theorem, which relates to the sides of a right angle triangle.

In our exercise, the Pythagorean identity helps to determine the missing sine or cosine values when one is known. For example, knowing \(\cos a = \frac{4}{5}\), you can find \(\sin a\) by rearranging the identity as \(\sin a = \sqrt{1 - \cos^2 a}\), selecting the positive square root because angle \(a\) lies in the first quadrant where sine is positive.

Trigonometric Function Values

The trigonometric function values for sine and cosine are based on the position of angles on the unit circle. These values can be either positive or negative, depending on the quadrant in which the angle lies.

Interpreting the Function Values

In the first quadrant, both sine and cosine values are positive, while in the fourth quadrant, cosine values are positive but sine values are negative. It is crucial for solving trigonometric expressions to remember these signs and apply them correctly, as seen in the step-by-step solution for \(\cos (a-b)\) when determining the signs for \(\sin a\) and \(\cos b\).

Angle Measurement in Radians

Angles can be measured in degrees or radians, and in mathematics, especially calculus, radian measure is the standard. A radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

The range of angles provided in the exercise, \(0