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Magazine Chapter 10: Problem 22
If \(\alpha\) is a Quadrant IV angle with \(\cos (\alpha)=\frac{\sqrt{5}}{5}\), and \(\sin (\beta)=\frac{\sqrt{10}}{10}\), where \(\frac{\pi}{2}<\beta<\pi\), find (a) \(\cos (\alpha+\beta)\) (b) \(\sin (\alpha+\beta)\) (c) \(\tan (\alpha+\beta)\) (d) \(\cos (\alpha-\beta)\) (e) \(\sin (\alpha-\beta)\) (f) \(\tan (\alpha-\beta)\)
Short Answer
Expert verified
(a) \(-\frac{\sqrt{2}}{10}\), (b) \(\frac{7\sqrt{2}}{10}\), (c) \(-7\), (d) \(-\frac{\sqrt{2}}{10}\), (e) \(\frac{\sqrt{2}}{2}\), (f) \(-5\).
Step by step solution
01
Determine the Sine of Angle 伪
Since \( \alpha \) is in Quadrant IV, \( \sin(\alpha) \) will be negative. Using \( \sin^2(\alpha) + \cos^2(\alpha) = 1 \), solve for \( \sin(\alpha) \):\[ \sin^2(\alpha) = 1 - \cos^2(\alpha) = 1 - \left( \frac{\sqrt{5}}{5} \right)^2 = 1 - \frac{5}{25} = \frac{20}{25} = \frac{4}{5} \]Therefore, \( \sin(\alpha) = -\sqrt{\frac{4}{5}} = -\frac{2\sqrt{5}}{5} \).
02
Determine the Cosine of Angle 尾
Since \( \frac{\pi}{2} < \beta < \pi \), \( \beta \) is in Quadrant II and \( \cos(\beta) \) will be negative. Using \( \sin^2(\beta) + \cos^2(\beta) = 1 \), solve for \( \cos(\beta) \):\[ \cos^2(\beta) = 1 - \sin^2(\beta) = 1 - \left( \frac{\sqrt{10}}{10} \right)^2 = 1 - \frac{10}{100} = \frac{90}{100} = \frac{9}{10} \]Therefore, \( \cos(\beta) = -\sqrt{\frac{9}{10}} = -\frac{3\sqrt{10}}{10} \).
03
Compute \( \cos(\alpha + \beta) \)
Use the angle sum formula for cosine: \[ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \]Substitute the known values:\[ \cos(\alpha + \beta) = \frac{\sqrt{5}}{5} \left(-\frac{3\sqrt{10}}{10} \right) - \left(-\frac{2\sqrt{5}}{5} \right) \left( \frac{\sqrt{10}}{10} \right) \]\[ = -\frac{3\sqrt{50}}{50} + \frac{2\sqrt{50}}{50} = -\frac{15\sqrt{2}}{50} + \frac{10\sqrt{2}}{50} = -\frac{5\sqrt{2}}{50} = -\frac{\sqrt{2}}{10} \]
04
Compute \( \sin(\alpha + \beta) \)
Use the angle sum formula for sine:\[ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \]Substitute the values:\[ \sin(\alpha + \beta) = \left(-\frac{2\sqrt{5}}{5} \right) \left(-\frac{3\sqrt{10}}{10} \right) + \left( \frac{\sqrt{5}}{5} \right) \left( \frac{\sqrt{10}}{10} \right) \]\[ = \frac{6\sqrt{50}}{50} + \frac{\sqrt{50}}{50} = \frac{7\sqrt{2}}{10} \]
05
Compute \( \tan(\alpha + \beta) \)
Use the formula for tangent of sum:\[ \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \]Substitute the previously found values:\[ \tan(\alpha + \beta) = \frac{\frac{7\sqrt{2}}{10}}{-\frac{\sqrt{2}}{10}} = -7 \]
06
Compute \( \cos(\alpha - \beta) \)
Use the angle difference formula for cosine:\[ \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \]Substitute known values:\[ \cos(\alpha - \beta) = \frac{\sqrt{5}}{5} \left(-\frac{3\sqrt{10}}{10} \right) + \left(-\frac{2\sqrt{5}}{5} \right) \left( \frac{\sqrt{10}}{10} \right) \]\[ = -\frac{3\sqrt{50}}{50} - \frac{2\sqrt{50}}{50} = -\frac{5\sqrt{2}}{10} = -\frac{\sqrt{2}}{10} \]
07
Compute \( \sin(\alpha - \beta) \)
Use the angle difference formula for sine:\[ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \]Substitute the values:\[ \sin(\alpha - \beta) = \left(-\frac{2\sqrt{5}}{5} \right) \left(-\frac{3\sqrt{10}}{10} \right) - \left( \frac{\sqrt{5}}{5} \right) \left( \frac{\sqrt{10}}{10} \right) \]\[ = \frac{6\sqrt{50}}{50} - \frac{\sqrt{50}}{50} = \frac{5\sqrt{2}}{10} = \frac{\sqrt{2}}{2} \]
08
Compute \( \tan(\alpha - \beta) \)
Use the formula for tangent of difference:\[ \tan(\alpha - \beta) = \frac{\sin(\alpha - \beta)}{\cos(\alpha - \beta)} \]Substitute the previously found values:\[ \tan(\alpha - \beta) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{10}} = -5 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle Sum and Difference Identities
The angle sum and difference identities are essential tools when solving problems involving trigonometric functions. These identities help find the sine, cosine, or tangent of the sum or difference of two angles based on the trigonometric values of individual angles. For cosine, the angle sum identity is:- \( \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \)And for sine, the identity is:- \( \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \)These formulas change slightly for angle differences:- \( \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \)- \( \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \)These identities are beneficial because they break down complex expressions into manageable parts. By knowing the sine and cosine of individual angles, you can calculate these trigonometric values for the combined angles easily.
Quadrant Angles
Understanding quadrant angles is crucial when working with trigonometric functions, especially when calculating sine and cosine for specific angles. The unit circle approach divides angles into four quadrants:
- Quadrant I: where all trigonometric values鈥攕ine, cosine, and tangent鈥攁re positive.
- Quadrant II: where sine is positive, but cosine and tangent are negative.
- Quadrant III: where tangent is positive, but sine and cosine are negative.
- Quadrant IV: where cosine is positive, and sine and tangent are negative.
Trigonometric Functions
Trigonometric functions are foundational in understanding angles and their relationships. There are three primary trigonometric functions: sine, cosine, and tangent:
- Sine, written as \( \sin \), is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
- Cosine, written as \( \cos \), is the ratio of the length of the adjacent side to the hypotenuse.
- Tangent, written as \( \tan \), is the ratio of the sine to the cosine or, equivalently, the opposite side to the adjacent side.
Sine and Cosine Relationships
The sine and cosine functions hold significant relationships with each other, mainly through trigonometric identities. Two of the crucial ones are the Pythagorean identity and the complementary angle identity.- The Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \) is utilized to find one trigonometric ratio when the other is known. For example, if \( \cos(\alpha) \) is given, \( \sin(\alpha) \) can be calculated using this identity.
Example: \( \sin^2(\alpha) = 1 - \cos^2(\alpha) \)- The complementary angle identity: It shows that the sine of an angle equals the cosine of its complement (and vice versa), formally described as \( \sin(90^\circ - x) = \cos(x) \).These relationships underpin our problem-solving strategies in trigonometry, ensuring we approach calculations with logical steps based on established mathematical rules.
Example: \( \sin^2(\alpha) = 1 - \cos^2(\alpha) \)- The complementary angle identity: It shows that the sine of an angle equals the cosine of its complement (and vice versa), formally described as \( \sin(90^\circ - x) = \cos(x) \).These relationships underpin our problem-solving strategies in trigonometry, ensuring we approach calculations with logical steps based on established mathematical rules.
