Trigonometry: Solving Right Triangles

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Questions and Answers

What is a necessary component for solving a right triangle?

  • Knowing the length of the hypotenuse
  • Knowing only one side
  • Knowing all three angles
  • Knowing the measures of two sides (correct)

Which function is primarily used in solving right triangles?

  • Algebraic functions
  • Trigonometric functions (correct)
  • Logarithmic functions
  • Exponential functions

When naming a right triangle, which letter represents the right angle?

  • The middle letter (correct)
  • The last letter
  • The first letter
  • Any of the three letters

What is required to find the values of unknowns in a right triangle using a calculator?

<p>The calculator must be scientific (C)</p> Signup and view all the answers

How can angles of elevation and depression be characterized in relation to right triangles?

<p>They depend on the position of the observer (D)</p> Signup and view all the answers

Which of the following statements is correct about acute angles in a right triangle?

<p>The sum of the acute angles is 90 degrees (C)</p> Signup and view all the answers

What is the process called when finding all measures of a triangle?

<p>Solving the triangle (A)</p> Signup and view all the answers

Which of the following is NOT a typical part involved in solving right triangles?

<p>Calculating the area (D)</p> Signup and view all the answers

What is the angle of elevation?

<p>The angle between a horizontal line and the line of sight to an object above the observer (D)</p> Signup and view all the answers

If the height of an object is greater than the observer's height, which angle is formed?

<p>Angle of elevation (C)</p> Signup and view all the answers

In the situation where Jay is flying a kite, how is the length of the string determined?

<p>By using the height of the kite and the angle the string makes with the ground (D)</p> Signup and view all the answers

What information is needed to calculate the angle of elevation of the sun when given a pole and its shadow?

<p>The height of the pole and the length of its shadow (A)</p> Signup and view all the answers

When Hannah looks at a cat on the ground from the barn, what type of angle is she observing?

<p>Angle of depression (C)</p> Signup and view all the answers

What is the relationship between sine and cosine for angles A and B in a right triangle?

<p>sin A = cos B (A), cos A = sin B (D)</p> Signup and view all the answers

How can the horizontal distance from a tower be calculated if the height of the tower and the angle subtended are known?

<p>Using the tangent function of the angle and the height (B)</p> Signup and view all the answers

In a right triangle, if angle A measures 30 degrees, what can be concluded about angle B?

<p>Angle B must be 60 degrees. (D)</p> Signup and view all the answers

In solving for the hypotenuse of a right triangle where angle A is known, which trigonometric ratio would commonly be used?

<p>Sine (D)</p> Signup and view all the answers

When using the Pythagorean theorem in a right triangle, which formula is used?

<p>a^2 + b^2 = c^2 (D)</p> Signup and view all the answers

If the length of side a in right triangle ACB is 24.07 cm, what method can be used to find side c?

<p>Employing the Pythagorean theorem (A)</p> Signup and view all the answers

When observing an object positioned below the observer's horizontal line of sight, which angle is identified?

<p>Angle of depression (B)</p> Signup and view all the answers

How can the angle of elevation and angle of depression be defined in terms of right triangles?

<p>They describe the angles formed by horizontal and vertical lines. (B)</p> Signup and view all the answers

If the length of side b in triangle ACB is unknown, which trigonometric function could be most useful if angle A is known?

<p>tan A (B)</p> Signup and view all the answers

Which of the following ratios represents the tangent of angle A in right triangle ACB?

<p>side a/side b (B)</p> Signup and view all the answers

To find the unknown length of side a at angle A in right triangle ACB, which relationship can be established?

<p>sin A = a/c (C)</p> Signup and view all the answers

Flashcards

Solving a right triangle

Finding the measures of all angles and sides of a right triangle.

Right triangle

A triangle with one angle measuring 90 degrees.

Acute Angle

An angle less than 90 degrees.

Solving Right Triangles

Determining all side lengths & angles of a right triangle, knowing some values.

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Trigonometric Functions

Functions in math relating ratios of sides of a right triangle to its angles.

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Angle of Elevation

Angle between the horizontal line and the upward line of sight from an observer to an object.

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Angle of Depression

Angle formed between a horizontal line and an observer's downward line of sight to an object.

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Scientific Calculator

Calculator used for calculations involving scientific equations, including trigonometric functions.

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Pythagorean theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

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Sine (sin)

Opposite side divided by hypotenuse.

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Cosine (cos)

Adjacent side divided by hypotenuse.

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Tangent (tan)

Opposite side divided by adjacent side.

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Line of Sight

The imaginary line from the observer's eye to the object being observed.

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Right Triangle Application

Using trigonometric ratios (sine, cosine, tangent) to solve for unknown sides or angles in a right triangle.

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Trigonometric Ratios

Relationships between the sides and angles in a right triangle. Include Sine, Cosine and Tangent ratios.

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Horizontal Distance

The distance between two points along a horizontal plane.

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Angle of Elevation Example

Kite string forms an angle with a horizontal surface to describe height.

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Angle of Elevation Problem

Calculate the angle between the horizontal and the line of sight given the height and shadow length of a pole.

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Study Notes

Right Triangle Solution

  • Students are expected to solve for unknown parts of a right triangle given known parts.
  • Apply trigonometric functions in problem situations involving right triangles.
  • Use a scientific calculator correctly to find unknown values.
  • Solve problems involving angles of elevation and depression.

Review Problems

  • Find the exact value of the following expressions, all given in square root format, for example √2/2

Solving Right Triangles

  • A right triangle is labeled with the sides as a, b and c (hypotenuse), and the angles as A, B, and C (right angle).
  • To solve a right triangle, find the measures of all the angles and sides.
  • Knowing two sides or a side and an acute angle allows solution.

Definitions of Trigonometric Functions

  • sin A = opposite/hypotenuse
  • cos A = adjacent/hypotenuse
  • tan A = opposite/adjacent
  • sin B = opposite/hypotenuse
  • cos B = adjacent/hypotenuse
  • tan B = opposite/adjacent

Pythagorean Theorem

  • a² + b² = c²

Example Problems

  • Example 1: Solve triangle ACB with B = 36°30′ and c = 72.4 cm.

  • The provided example includes sides calculated to the nearest hundredth ( cm, or centimeters).

    • A = 53°30′
    • a ≈ 58.20 cm
    • b ≈ 43.06 cm

  • Example 2: Find unknown parts of triangle ACB if A = 24°45′ and a = 24.07 cm.

    • B ≈ 65°15′
    • c ≈ 57.49 cm
    • b ≈ 52.21 cm

  • Example 3: Solve triangle ACB with c = 201.25 cm, and B = 42°20'33.38''.

    • A = 47°39'26.62''
    • b ≈ 135.55 cm
    • a ≈ 148.75 cm

Applications of Right Triangles

  • Application scenarios involve angles of elevation and depression.

Angle of Elevation and Depression

  • Angle of elevation: the angle from the horizontal to a point above an observer.
  • Angle of depression: the angle from the horizontal to a point below an observer.

Example Application Problem:

  • Example 1: Jay is flying a kite. The string of the kite makes an angle of θ with the ground.

    • If the height of the kite is 21 cm, find the length of the string Jay used.
    • Answer: 31.08 cm

  • Example 2: A pole 60 ft high casts a 48-ft shadow.

  • Find the angle of elevation of the sun.

  • Answer:

  • Calculation needed to solve for the angle.

  • Example 3: Hannah is on top of a 25 ft tall barn.

  • She sees a cat on the ground below.

  • The angle of depression is given(angle).

  • Calculate how many feet the cat needs to walk to reach the barn.

  • Answer: 29.74 ft

Practice Exercises

  • Exercise 1: Solve the right triangle where c = 37.2 and A = 34°10′.
  • Exercise 2: Solve the right triangle where a = 41.72 and b = 57.34.
  • Exercise 3: Jun is walking along a road. The top of a 35-meter tower subtends an angle with the ground. Find Jun's horizontal distance from the tower.

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Solutions Of Right Triangles PDF

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