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Questions and Answers
A square's diagonal has a length of $7\sqrt{2}$. What is the area of the square?
A square's diagonal has a length of $7\sqrt{2}$. What is the area of the square?
- 49 (correct)
- 7
- 98
- 14
In a 30-60-90 triangle, the shortest side has a length of 5. What is the length of the hypotenuse?
In a 30-60-90 triangle, the shortest side has a length of 5. What is the length of the hypotenuse?
- 10 (correct)
- $\frac{5\sqrt{3}}{2}$
- $\frac{10\sqrt{3}}{3}$
- 5\sqrt{3}
Given $\sin(\theta) = \frac{\sqrt{3}}{2}$ and $\cos(\theta) = \frac{1}{2}$, what is the value of $\tan(\theta)$?
Given $\sin(\theta) = \frac{\sqrt{3}}{2}$ and $\cos(\theta) = \frac{1}{2}$, what is the value of $\tan(\theta)$?
- 2
- $\sqrt{3}$ (correct)
- $\frac{\sqrt{3}}{3}$
- 1
An equilateral triangle has a side length of 8. What is the exact area of the triangle?
An equilateral triangle has a side length of 8. What is the exact area of the triangle?
If $\cos(A) = \frac{5}{13}$, what is the value of $\sin(A)$ if $A$ is an acute angle in a right triangle?
If $\cos(A) = \frac{5}{13}$, what is the value of $\sin(A)$ if $A$ is an acute angle in a right triangle?
What is the simplified form of $\frac{8}{\sqrt{2}}$?
What is the simplified form of $\frac{8}{\sqrt{2}}$?
In a right triangle, one angle measures 30°, and the hypotenuse has a length of 10. What is the length of the side opposite the 30° angle?
In a right triangle, one angle measures 30°, and the hypotenuse has a length of 10. What is the length of the side opposite the 30° angle?
If $\tan(\theta) = 1$, what is the measure of angle $\theta$ in degrees, assuming $0^\circ < \theta < 90^\circ$?
If $\tan(\theta) = 1$, what is the measure of angle $\theta$ in degrees, assuming $0^\circ < \theta < 90^\circ$?
An observer is standing 50 feet away from the base of a tree. The angle of elevation to the top of the tree is 60°. Approximately how tall is the tree?
An observer is standing 50 feet away from the base of a tree. The angle of elevation to the top of the tree is 60°. Approximately how tall is the tree?
What is the value of $x$ in the expression $\frac{\sqrt{27x}}{\sqrt{3}} = 9$?
What is the value of $x$ in the expression $\frac{\sqrt{27x}}{\sqrt{3}} = 9$?
Flashcards
Simplify and Rationalize
Simplify and Rationalize
Simplifying involves reducing a fraction to its simplest form, while rationalizing removes radicals from the denominator.
45-45-90 Triangle Properties
45-45-90 Triangle Properties
45-45-90 triangles have sides in ratio x : x : x√2, where x is the length of each leg.
30-60-90 Triangle Properties
30-60-90 Triangle Properties
In a 30-60-90 triangle, sides are in ratio x : x√3 : 2x, where x is the shortest side.
Sine, Cosine, Tangent
Sine, Cosine, Tangent
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Area of a square given diagonal
Area of a square given diagonal
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Cosine Definition
Cosine Definition
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Sine Definition
Sine Definition
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Tangent Definition
Tangent Definition
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Study Notes
- Self-assessment used to identify the strengths and weaknesses.
- Practice problems used to review skills
Warm-Up Activity Skills
- Properties of 45-45-90 triangles to find side lengths
- Properties of 30-60-90 triangles to find side lengths
- Simplify and rationalize answers
- Estimate angle measures using trig ratios.
- Calculate side lengths using cosine, sine, and tangent
Simplify and Rationalize Expressions
- Examples simplifying and rationalizing expressions
- Example a: 11/√5 simplified to (11√5)/5
- Example b: 7/(2√2) simplified to (7√2)/4
- Example c: (6/√3) simplified to 6√2
- Example d: 3/√21 simplified to (√21)/7
Square's Diagonal and Area
- Square's diagonal is 15.
- Area of the square = x² = (15/√2)² = 225/2 = 112.5 units²
Triangle: Find x and y
- Find the value of variables x and y in the triangle
- 4 = x√2, so x = 4/√2 = 2√2; since it's a 45-45-90 triangle, x = y = 2√2
Triangle: Find a and b
- b = 2√2
- a = x√2 = 2√2 * √2 = 2 * 2 = 4
Find the Value of x and y in the Triangle
- Solve for x and y
- 12 = y√3, which gives y = 12/√3 = 4√3
- x = 2y = 2 * 4√3 = 8√3
Equilateral Triangle
- Equilateral triangle with side length of 12
- Area calculation = bh = 6 * 6√3 = 36√3 units²
Solving for Variables in Diagrams
- Diagram (a): x = 5√6 / 3
- Diagram (b): x = 2√6
- Diagram (c): x = 18√2
- Diagram (d): x = 10
Estimating Trigonometric Values
- Includes a table to estimate the values in order
- Value A: adjacent leg/hypotenuse, opposite leg/hypotenuse, opposite leg/adjacent leg
- Value B: adjacent leg/hypotenuse, opposite leg/hypotenuse, opposite leg/adjacent leg
Finding Hypotenuse Length
- A right triangle, one angle is 50°, adjacent leg is 12 cm
- cos(50°) = 12/x, which gives x ≈ 18.67 cm
Determining Opposite Length
- Right triangle where one angle is 60°, adjacent leg is 16 cm
- The length of the opposite is 16√3
Estimating Angle Measures
- sin(θ) = 0.7245, θ is between 40° and 50°
- cos(θ) = 0.9659, θ is between 10° and 20°
- Tan(θ) = 3.7321, θ is between 70° and 80°
- sin(θ) = 0.500, θ is 30°
Trigonometric Ratios
- sin(J) = √2 / 3
- cos(J) = 2√2 / 3
- tan(J) = √2 / 4
Solving for Missing Sides
- cos(36°) = x/7 which implies x = 5.7 units
- sin(36°) = y/7 which implies y = 4.1 units
Solving for Missing Sides in Diagram
- cos(12.7°) = 40/x which implies x = 41.0 units
- tan(12.7°) = y/40 which implies y = 9.0 units
Calculating Distance
- cos(51°) = A/8
- Distance A ≈ 5.03 feet
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