Solving for Tangents of Sums and Differences Given Sine and Cosine Values
In trigonometry, we often need to find the tangent of the sum or difference of two angles given the sine and cosine values of the individual angles. This problem demonstrates how to solve for tan(A B) and tan(A - B) given that sin A 3/5 and cos B -5/13. This article will guide you through the process, breaking down each step to ensure clarity and understanding.
Understanding the Problem and the Formulas
To solve for the tangents of the sum and difference of angles, we use the following formulas:
Tan of the sum formula: (tan(A B) frac{tan A tan B}{1 - tan A tan B}) Tan of the difference formula: (tan(A - B) frac{tan A - tan B}{1 tan A tan B})Our goal is to find tan A B and tan A - B using the given values sin A 3/5 and cos B -5/13.
Step 1: Calculate tan A
Given sin A 3/5, we can find cos A using the Pythagorean identity:
(cos^2 A 1 - sin^2 A 1 - left(frac{3}{5}right)^2 1 - frac{9}{25} frac{16}{25})
Therefore, (cos A sqrt{frac{16}{25}} frac{4}{5}) (assuming A is in the first quadrant). Now we can find tan A as follows:
(tan A frac{sin A}{cos A} frac{frac{3}{5}}{frac{4}{5}} frac{3}{4})
Step 2: Calculate tan B
Given cos B -5/13, we can find sin B using the Pythagorean identity:
(sin^2 B 1 - cos^2 B 1 - left(-frac{5}{13}right)^2 1 - frac{25}{169} frac{144}{169})
Thus, (sin B sqrt{frac{144}{169}} frac{12}{13}) (assuming B is in the second quadrant). Now we can find tan B as follows:
(tan B frac{sin B}{cos B} frac{frac{12}{13}}{-frac{5}{13}} -frac{12}{5})
Step 3: Calculate tan(A B)
Using the sum formula, we substitute the values of tan A and tan B into the formula:
(tan(A B) frac{tan A tan B}{1 - tan A tan B} frac{frac{3}{4} - frac{12}{5}}{1 - left(frac{3}{4}right)left(-frac{12}{5}right)})
Calculating the numerator:
(frac{3}{4} - frac{12}{5} frac{15}{20} - frac{48}{20} -frac{33}{20})
Calculating the denominator:
(1 - left(frac{3}{4}right)left(-frac{12}{5}right) 1 frac{36}{20} frac{56}{20} frac{14}{5})
Therefore, (tan(A B) -frac{33}{20} cdot frac{5}{14} -frac{33}{56})
Step 4: Calculate tan(A - B)
Using the difference formula, we substitute the values of tan A and tan B into the formula:
(tan(A - B) frac{tan A - tan B}{1 tan A tan B} frac{frac{3}{4} frac{12}{5}}{1 left(frac{3}{4}right)left(-frac{12}{5}right)})
Calculating the numerator:
(frac{3}{4} frac{12}{5} frac{15}{20} frac{48}{20} frac{63}{20})
Calculating the denominator:
(1 left(frac{3}{4}right)left(-frac{12}{5}right) 1 - frac{36}{20} -frac{16}{20} -frac{4}{5})
Therefore, (tan(A - B) frac{frac{63}{20}}{-frac{4}{5}} -frac{63}{16})
Final Results
The final results are:
tan(A B) -33/56 tan(A - B) -63/16Summary
By following the steps outlined in this article, you can solve for the tangents of the sum and difference of angles given the sine and cosine values of the individual angles. The process involves using the Pythagorean identity to find the cosine or sine values and the tangent sum and difference formulas to determine the required tangents.