For Exercises 33-40, find the exact value. 40. tan arccos arcsin |-
Trigonometric Identities
Trigonometry in mathematics deals with the right-angled triangle’s angles and sides. By trigonometric identities, we mean the identities we use whenever we need to express the various trigonometric functions in terms of an equation.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverse of normal trigonometric functions. Alternatively denoted as cyclometric or arcus functions, these inverse trigonometric functions exist to counter the basic trigonometric functions, such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). When trigonometric ratios are calculated, the angular values can be calculated with the help of the inverse trigonometric functions.
![**Exercise Problem:**
For Exercises 33–40, find the exact value. (See Example 4)
**Problem 40:**
\[ \text{tan} \left( \arccos \frac{1}{5} - \arcsin \frac{3}{5} \right) \]
In this exercise, you are required to find the exact value of the tangent of the difference between the arc cosine of \(\frac{1}{5}\) and the arc sine of \(\frac{3}{5}\).
### Steps to Solve:
1. **Understand Definitions:**
- \(\arccos(x)\) gives the angle whose cosine is \(x\).
- \(\arcsin(x)\) gives the angle whose sine is \(x\).
2. **Use Right Triangle Relationships:**
- For \(\arccos \frac{1}{5}\), let's designate this angle as \(\theta_1\). Therefore, \(\cos \theta_1 = \frac{1}{5}\).
- Similarly, for \(\arcsin \frac{3}{5}\), let's designate this angle as \(\theta_2\). Therefore, \(\sin \theta_2 = \frac{3}{5}\).
3. **Using Pythagorean Identity:**
- Determine \(\sin \theta_1\): Knowing \(\cos^2 \theta_1 + \sin^2 \theta_1 = 1\),
\[
\sin \theta_1 = \sqrt{1 - \left(\frac{1}{5}\right)^2} = \sqrt{1 - \frac{1}{25}} = \sqrt{\frac{24}{25}} = \frac{2\sqrt{6}}{5}
\]
- Determine \(\cos \theta_2\): Knowing \(\sin^2 \theta_2 + \cos^2 \theta_2 = 1\),
\[
\cos \theta_2 = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}
\]
4. **Angle Difference Identity for Tangent:**
- The difference of angles \(\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F231fae22-ac62-4382-9eb0-95d912af77d1%2F856e9cf1-b80c-45bf-8ae3-6437c0fc0e3b%2Fd9wqfff.jpeg&w=3840&q=75)
Step by step
Solved in 6 steps with 12 images
