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.
![### Problem: Finding the Sine of ∠F in a Right Triangle
#### Given:
- Right triangle \( \triangle GHF \)
- \( \overline{GH} \) = 24 (opposite side to ∠F)
- \( \overline{HF} \) = 10 (adjacent side to ∠F)
- \( \overline{GF} \) = 26 (hypotenuse)
### Step-by-Step Solution:
1. **Identify the Trigonometric Function**:
The sine (sin) of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
\[
\sin(\angle F) = \frac{\text{Opposite side}}{\text{Hypotenuse}}
\]
2. **Plug in Known Values**:
Here, the opposite side to ∠F is \( \overline{GH} \) and the hypotenuse is \( \overline{GF} \):
\[
\sin(F) = \frac{\overline{GH}}{\overline{GF}} = \frac{24}{26}
\]
3. **Simplify the Fraction**:
Simplify the fraction \( \frac{24}{26} \) by finding the greatest common divisor (GCD) of 24 and 26, which is 2:
\[
\sin(F) = \frac{24 \div 2}{26 \div 2} = \frac{12}{13}
\]
4. **Final Answer**:
Therefore, the sine of ∠F is \( \frac{12}{13} \).
### Graphical Representation:
- The diagram represents a right triangle \( \triangle GHF \) with a right angle at ∠H.
- The side \( \overline{GH} \) (24 units) is vertical and opposite to ∠F.
- The side \( \overline{HF} \) (10 units) is horizontal and adjacent to ∠F.
- The hypotenuse \( \overline{GF} \) is 26 units.
### Answer Box:
- Input box is provided for simplifying the fraction and writing it as a proper fraction, improper fraction, or whole number.
\[
\sin(F) = \frac{12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fedeaf9eb-5dba-4b61-bd52-fd0e4f5aa908%2F91abaf40-9b98-4958-bee1-9d069cd24a3e%2Fw88nq3l_processed.png&w=3840&q=75)
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