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Questions and Answers
If $\sin(A) = \frac{2}{5}$ and the hypotenuse is 5 units, what is the length of the opposite side?
If $\sin(A) = \frac{2}{5}$ and the hypotenuse is 5 units, what is the length of the opposite side?
2 units
In a right triangle, if the adjacent side is 3 units and the hypotenuse is 5 units, what is $\cos(A)$?
In a right triangle, if the adjacent side is 3 units and the hypotenuse is 5 units, what is $\cos(A)$?
$\frac{3}{5}$
If $\tan(A) = \frac{3}{4}$, and the opposite side is 3 units, what is the length of the adjacent side?
If $\tan(A) = \frac{3}{4}$, and the opposite side is 3 units, what is the length of the adjacent side?
4 units
If $\sin(A) = \frac{3}{5}$ and $\cos(A) = \frac{4}{5}$, verify that $\sin^2(A) + \cos^2(A) = 1$.
If $\sin(A) = \frac{3}{5}$ and $\cos(A) = \frac{4}{5}$, verify that $\sin^2(A) + \cos^2(A) = 1$.
In a 30-60-90 triangle, if the short leg is 2 units, what is the length of the hypotenuse?
In a 30-60-90 triangle, if the short leg is 2 units, what is the length of the hypotenuse?
Simplify the expression $\sqrt{18} + \sqrt{50}$.
Simplify the expression $\sqrt{18} + \sqrt{50}$.
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Study Notes
Trigonometric Ratios
Sine (sin)
- Opposite side / Hypotenuse
- sin(A) = opposite side / hypotenuse
- Example: sin(A) = 3/5, if the opposite side is 3 units and the hypotenuse is 5 units
Cosine (cos)
- Adjacent side / Hypotenuse
- cos(A) = adjacent side / hypotenuse
- Example: cos(A) = 4/5, if the adjacent side is 4 units and the hypotenuse is 5 units
Tangent (tan)
- Opposite side / Adjacent side
- tan(A) = opposite side / adjacent side
- Example: tan(A) = 3/4, if the opposite side is 3 units and the adjacent side is 4 units
Pythagorean Identity
- sin^2(A) + cos^2(A) = 1
- Used to find the length of a side in a right triangle
Special Right Triangles
30-60-90 Triangles
- Hypotenuse: 2 units
- Short leg: 1 unit
- Long leg: √3 units
45-45-90 Triangles
- Hypotenuse: √2 units
- Legs: 1 unit each
Simplifying Radicals
Rules
- √(a × b) = √a × √b
- √(a / b) = √a / √b
- a√b = √(a^2 × b)
Examples
- Simplify √(8)
- √(8) = √(4 × 2) = √4 × √2 = 2√2
- Simplify 3√(12)
- 3√(12) = 3√(4 × 3) = 3 × 2√3 = 6√3
Trigonometric Ratios
- Sine (sin) is the ratio of the opposite side to the hypotenuse
- Cosine (cos) is the ratio of the adjacent side to the hypotenuse
- Tangent (tan) is the ratio of the opposite side to the adjacent side
Examples of Trigonometric Ratios
- sin(A) = opposite side / hypotenuse (example: sin(A) = 3/5)
- cos(A) = adjacent side / hypotenuse (example: cos(A) = 4/5)
- tan(A) = opposite side / adjacent side (example: tan(A) = 3/4)
Pythagorean Identity
- sin^2(A) + cos^2(A) = 1
- Used to find the length of a side in a right triangle
Special Right Triangles
30-60-90 Triangles
- Hypotenuse: 2 units
- Short leg: 1 unit
- Long leg: √3 units
45-45-90 Triangles
- Hypotenuse: √2 units
- Legs: 1 unit each
Simplifying Radicals
Rules
- √(a × b) = √a × √b
- √(a / b) = √a / √b
- a√b = √(a^2 × b)
Examples of Simplifying Radicals
- √(8) + √(8) = √(4 × 2) = √4 × √2 = 2√2
- 3√(12) + 3√(12) = 3√(4 × 3) = 3 × 2√3 = 6√3
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