Question
Give all the formulas for calculating right angle triangle's side and angle and non-right angle triangle's side and angle and some of the tricks or pattern to remember the formula
Answers
GPT-4o mini
Certainly! Below are the key formulas for calculating the sides and angles in both right triangles and non-right triangles, along with some mnemonic devices to help remember them.
### Right Triangle Formulas
For a right triangle with sides \( a \) and \( b \) (the legs) and hypotenuse \( c \):
1. **Pythagorean Theorem**:
\[
c^2 = a^2 + b^2
\]
- **Mnemonic**: "The square of the hypotenuse is the sum of the squares of the other two sides."
2. **Trigonometric Ratios**:
- Sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
- Cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- Tangent: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
- **Mnemonic**: "SOH-CAH-TOA"
- Sine = SOH (Sine = Opposite / Hypotenuse)
- Cosine = CAH (Cosine = Adjacent / Hypotenuse)
- Tangent = TOA (Tangent = Opposite / Adjacent)
3. **Finding Angles**:
- To find angles using the inverses:
- \( \theta = \sin^{-1} \left(\frac{a}{c}\right) \)
- \( \theta = \cos^{-1} \left(\frac{b}{c}\right) \)
- \( \theta = \tan^{-1} \left(\frac{a}{b}\right) \)
### Non-Right Triangle Formulas
For any triangle (not necessarily right-angled), you can use the following:
1. **Law of Sines**:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
- Where \( A, B, C \) are the angles opposite to sides \( a, b, c \) respectively.
- **Mnemonic**: "Sine's ratio, keep it in line."
2. **Law of Cosines**:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
- This can be rearranged for any side. For example:
\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]
- **Mnemonic**: "Square, square, less double product times cosine."
3. **Finding Angles Using Law of Cosines**:
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
- **Mnemonic**: "Take the opposite side squared, and rearrange the rest."
### Additional Patterns to Remember
- **Pythagorean Triples**: Some common sets that satisfy \( a^2 + b^2 = c^2 \) are:
- (3, 4, 5)
- (5, 12, 13)
- (8, 15, 17)
- **Special Right Triangles**:
- 45°-45°-90° triangle: \( 1:1:\sqrt{2} \)
- 30°-60°-90° triangle: \( 1:\sqrt{3}:2 \)
### Summary
- For right triangles, use the Pythagorean theorem and trigonometric ratios (SOH-CAH-TOA).
- For non-right triangles, use the Law of Sines and Law of Cosines.
- Remember the special triangles and Pythagorean triples for quick calculations.
These formulas and mnemonics can provide a solid basis for solving problems involving triangles!
### Right Triangle Formulas
For a right triangle with sides \( a \) and \( b \) (the legs) and hypotenuse \( c \):
1. **Pythagorean Theorem**:
\[
c^2 = a^2 + b^2
\]
- **Mnemonic**: "The square of the hypotenuse is the sum of the squares of the other two sides."
2. **Trigonometric Ratios**:
- Sine: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
- Cosine: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- Tangent: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
- **Mnemonic**: "SOH-CAH-TOA"
- Sine = SOH (Sine = Opposite / Hypotenuse)
- Cosine = CAH (Cosine = Adjacent / Hypotenuse)
- Tangent = TOA (Tangent = Opposite / Adjacent)
3. **Finding Angles**:
- To find angles using the inverses:
- \( \theta = \sin^{-1} \left(\frac{a}{c}\right) \)
- \( \theta = \cos^{-1} \left(\frac{b}{c}\right) \)
- \( \theta = \tan^{-1} \left(\frac{a}{b}\right) \)
### Non-Right Triangle Formulas
For any triangle (not necessarily right-angled), you can use the following:
1. **Law of Sines**:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]
- Where \( A, B, C \) are the angles opposite to sides \( a, b, c \) respectively.
- **Mnemonic**: "Sine's ratio, keep it in line."
2. **Law of Cosines**:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
- This can be rearranged for any side. For example:
\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]
- **Mnemonic**: "Square, square, less double product times cosine."
3. **Finding Angles Using Law of Cosines**:
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]
- **Mnemonic**: "Take the opposite side squared, and rearrange the rest."
### Additional Patterns to Remember
- **Pythagorean Triples**: Some common sets that satisfy \( a^2 + b^2 = c^2 \) are:
- (3, 4, 5)
- (5, 12, 13)
- (8, 15, 17)
- **Special Right Triangles**:
- 45°-45°-90° triangle: \( 1:1:\sqrt{2} \)
- 30°-60°-90° triangle: \( 1:\sqrt{3}:2 \)
### Summary
- For right triangles, use the Pythagorean theorem and trigonometric ratios (SOH-CAH-TOA).
- For non-right triangles, use the Law of Sines and Law of Cosines.
- Remember the special triangles and Pythagorean triples for quick calculations.
These formulas and mnemonics can provide a solid basis for solving problems involving triangles!