Unit 7: Trigonometry

MPM2D Unit 7: Trigonometry

Lesson 7.0: Similar Triangles

Learning Goals:

• Understand the properties of similar triangles.

• Be able to apply similarity in triangles through different methods.

Definitions:

• Congruent Figures: Same shape, same size.

• Similar Figures: Same shape, different size.

Properties of Similar Triangles:

1. Similar triangles have the same angles.

2. The ratios of corresponding sides in similar triangles are equal.

3. The ratio of the areas of the triangles is equal to the ratio of the squares of the corresponding sides:
      $ext{Area Ratio} = rac{a^2}{b^2}$ (for corresponding sides a and b).
       

Proving Triangles are Similar:

Methods:

1. Angle-Angle Method (AA): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

2. Side-Angle-Side Method (SAS): If two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal, the triangles are similar.

3. Side-Side-Side Method (SSS): If the three sides of one triangle are proportional to the three sides of another triangle, the triangles are similar.

Examples:

• Example 1: (AA) Prove that triangles ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 are similar given angles are known.

• Example 2: (SAS) Prove that triangles ∆𝐴𝐵𝐶 and ∆𝑆𝑇𝑈 are similar by knowing certain sides and angle.

• Example 3: (SSS) Prove that triangles ∆𝐴𝐵𝐶 and ∆𝑋𝑌𝑍 are similar by comparing all sides.

Practice:

• Determine if given pairs of triangles are similar.

• Write similarity statements for similar triangles based on provided information.

Lesson 7.1: The Tangent Ratio

Learning Goals:

• Understand and apply the tangent ratio in right triangles.

Key Concepts:

• Solving a triangle involves finding all angles and side lengths.

• If two right triangles are similar, their side lengths are proportional.

The Tangent Ratio:

• Definition: The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side:
    $an( heta) = rac{ ext{Opposite}}{ ext{Adjacent}}$

Examples:

1. Example 1: Find side lengths given the angle 36.9° and adjacent side (e = 12 cm).

2. Example 2: Calculate $an(36.9°)$ using a calculator.

3. Draw and label right triangles to visualize relationships between angles and sides.

Lesson 7.2: The Sine and Cosine Ratios

Learning Goals:

• Understand sine and cosine ratios and how to use them to find unknowns in right triangles.

Key Concepts:

• Sine and cosine ratios can be applied to find lengths and angles in right triangles.

• The sine of an angle is the ratio of the length of the opposite side to the hypotenuse:
    $ext{sin}( heta) = rac{ ext{Opposite}}{ ext{Hypotenuse}}$

• The cosine of an angle is the ratio of the adjacent side to the hypotenuse:
    $ext{cos}( heta) = rac{ ext{Adjacent}}{ ext{Hypotenuse}}$

Examples:

1. Pythagorean Theorem to find missing sides.

2. Simplify sine and cosine ratios given sides.

3. Solve for angles using sine and cosine laws.

Lesson 7.3: Solving Problems Involving Right Triangles

Learning Goals:

• Apply trigonometric ratios to solve real-world problems connected to right triangles.

Examples:

1. Example 1: Finding length of kite string using angle and height.

2. Example 2: Finding the length of the hypotenuse given an angle and the adjacent side.

3. Example 3: Use of isosceles triangle properties to find height, perimeter, and area.

4. Example 4: Area calculation given base and height.

Practice Problems:

• Solve for dimensions based on provided angles and lengths.

Lesson 7.4: Solving Problems with Two Right Triangles

Learning Goals:

• Understand the interactions between two right triangles in different scenarios and apply trigonometry accordingly.

Examples:

1. Example 1: Analyzing a scenario involving angle of depression to find distances.

2. Example 2: Calculating distances and heights using elevation angles.

Lesson 7.5: The Sine Law

Learning Goals:

• Understand how to apply the Sine Law to non-right angle triangles.

Key Concepts:

• The Sine Law states that:
    $rac{a}{ ext{sin}(A)} = rac{b}{ ext{sin}(B)} = rac{c}{ ext{sin}(C)}$
    where a, b, c are the sides opposite angles A, B, and C respectively.

Derivation Scenarios:

1. When you know two angles and one side (AAS or ASA).

2. When you know two sides and an angle opposite one of those sides (SSA).

1. Given triangle ABC, if angle A = 30° and angle B = 60°, find side a if side b = 10 cm.

2. In triangle DEF, angles D = 45° and E = 75°, what is the length of side f if side e = 12 cm?

Lesson 7.6: The Cosine Law

Learning Goals:

• Understand how to apply the Cosine Law for more complex triangle calculations.

Key Concepts:

• The Cosine Law is expressed as:
    $c^2 = a^2 + b^2 - 2ab ext{cos}(C)$
    Allows the calculation of side lengths and angles in non-right triangles.

Scenarios:

• Use it when you know two sides and the included angle (SAS).

• Use when all three sides are known (SSS).

Lesson 7.7: Solving Problems Using Trigonometry

Learning Goals:

• Apply learned trig principles to various practical scenarios.

Examples:

1. Using a theodolite to measure angles and heights.

2. Determining building heights using angles of elevation and distances.

3. Real-life application problems involving direct measurements and angular calculations.

Unit Review

Textbook Review Assignments:

• Review materials from pages 333, 362, 373, and additional practice problems from Unit Booklet.

Test Preparation:

• Ensure proficiency on angles, sides, the law of sines, and the law of cosines.

Additional Practice and Homework Assignments:

• Complete exercises from specified pages in the textbook and unit booklet relating to similar triangles, tangent, sine, and cosine ratios, as well as problem-solving with two right triangles.