Understanding the Cosine of the Sum of Two Angles
When dealing with trigonometric functions, it's often necessary to find the cosine of the sum of two angles when only the sine and cosine of each angle are given. In this article, we will walk through a step-by-step process to find cosA - B, given that sinA 3/5 and cosB 12/13. Our goal is to illustrate the application of trigonometric identities and the cosine addition formula.
The Given Information and Initial Setup
Our given information is sinA 3/5 and cosB 12/13. We will use the cosine addition formula to find cosA - B.
Step 1: Find cosA
Using the Pythagorean identity, we start by finding cosA:
cos^2A sin^2A 1
Substituting sinA 3/5 into the identity:
cos^2A (3/5)^2 1
cos^2A 9/25 1
cos^2A 1 - 9/25
cos^2A 16/25
Since A is an acute angle, cosA 4/5.
Step 2: Find sinB
Similarly, we will find sinB using the Pythagorean identity:
sin^2B cos^2B 1
Substituting cosB 12/13 into the identity:
sin^2B (12/13)^2 1
sin^2B 144/169 1
sin^2B 1 - 144/169
sin^2B 25/169
Since B is an acute angle, sinB 5/13.
Step 3: Use the Cosine Addition Formula
Now we will substitute the values into the cosine addition formula:
cosA - B cosA * cosB - sinA * sinB
Substituting the values:
cosA - B (4/5) * (12/13) - (3/5) * (5/13)
Calculating each term:
cosA - B 48/65 - 15/65
cosA - B 33/65
Therefore, the value of cosA - B is:
cosA - B 33/65
Additional Insights and Applications
To further illustrate the use of these identities, let's break down the process step-by-step:
Using Trigonometric Identities
We can use the identity:
cos^2x - sin^2x 1
To find cosA and cosB:
Given sinA 3/5 and using the identity:
cos^2A 1 - sin^2A
cos^2A 1 - (3/5)^2
cos^2A 1 - 9/25
cos^2A 16/25
Since A is acute, cosA 4/5.
Similarly, for cosB 12/13:
Given sinB 5/13 and using the identity:
cos^2B 1 - sin^2B
cos^2B 1 - (5/13)^2
cos^2B 1 - 25/169
cos^2B 144/169
Since B is acute, cosB 12/13.
Substituting these values into the cosine addition formula:
cosA - B (4/5) * (12/13) - (3/5) * (5/13)
cosA - B 48/65 - 15/65
cosA - B 33/65
Conclusion
In conclusion, by using the cosine addition formula and trigonometric identities, we have successfully determined the value of cosA - B given the specific sine and cosine values. This process is a critical skill in trigonometry and a valuable tool in advanced mathematical and engineering applications.