Geometry and Trigonometry (IB)

1. Trigonometry of Right-Angled Triangles:

1.1. Definition of Trigonometric Ratios:

1.2. Pythagorean Theorem:

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

1.3. Triangle relationships:

1.4. Solving Right-Angled Triangles:

   - Use trigonometric ratios to find missing side lengths or angle measures in right-angled triangles.

   - Common methods include using SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent) and Pythagorean theorem.

1. 5. Special Right-Angled Triangles:

   - 45-45-90 Triangle: In an isosceles right-angled triangle, the two legs are congruent, and the ratio of the length of each leg to the length of the hypotenuse is

  - 30-60-90 Triangle: In a right-angled triangle with one acute angle measuring 30° and the other measuring 60°, the ratio of the lengths of the sides is

1.6. Applications in Geometry and Physics:

   - Trigonometric ratios are used extensively in geometry to find unknown lengths or angles in triangles.

   - In physics, trigonometry is used to analyze forces, motion, and trajectories, among other things.

1.7. Graphs of Trigonometric Functions:

   - The graphs of sine, cosine, and tangent functions exhibit periodic behavior and are used to model various phenomena in mathematics and physics.

1.8. Trigonometric Identities:

2. Sine Theorem 

2.1. Definition:

• The Sine Theorem, also known as the Law of Sines, is a fundamental trigonometric relationship applicable to triangles.

• It states that for any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant:

  

• This relationship holds for any triangle, whether it is acute, obtuse, or right-angled.

2.2. Conditions for Applicability:

   - The Sine Theorem is particularly useful when dealing with triangles where at least one side and its corresponding angle (side-angle pair) are known.

   - It is essential to understand the conditions for applying the theorem effectively, especially in solving triangles and determining the ambiguity of solutions.

2.3. Applications:

   - Solving Triangles: Given certain information about a triangle (e.g., side lengths and angles), the Sine Theorem can be applied to find unknown side lengths or angle measures.

   - Ambiguity in Solutions: Understanding the potential for two possible solutions in certain cases, particularly when dealing with ambiguous cases in solving triangles.

2.4. Derivation and Proof:

The Sine Theorem can be derived using the Law of Sines in trigonometry, which states that for any triangle ABC:

By cross-multiplying, we obtain the Sine Theorem:

2.5. Special Cases:

   - While the Sine Theorem applies to all triangles, discussing special cases such as right-angled triangles and equilateral triangles can deepen understanding and illustrate specific scenarios.

2.6. Example problems: 

   - Providing examples of problems where the Sine Theorem is used to solve triangles or analyze geometric relationships can enhance comprehension and application skills.

2.7. Graphical Representations:

   - Visual representations, such as diagrams and graphs, can aid in understanding the geometric relationships described by the Sine Theorem and its applications in solving triangles.

2.8. Real-world Contexts:

   - Exploring real-world contexts where the Sine Theorem is applicable, such as navigation, surveying, and engineering, can demonstrate its relevance and practical utility.

3. Cosine Theorem

3.1. Definition:

• The Cosine Theorem, also known as the Law of Cosines, is a fundamental trigonometric relationship applicable to triangles.

• It relates the lengths of the sides of a triangle to the cosine of one of its angles:

3.2. Conditions for Applicability:

• The Cosine Theorem is applicable to any triangle, whether it is acute, obtuse, or right-angled.

• It is particularly useful when dealing with triangles where the measures of two sides and the included angle (side-angle-side) are known.

3.3. Applications:

• Solving Triangles: Given certain information about a triangle (e.g., side lengths and included angle), the Cosine Theorem can be applied to find unknown side lengths or angle measures

• Ambiguity in solutions: Understanding the potential for two possible solutions in certain cases, particularly when dealing with ambiguous cases in solving triangles.

3.4. Special cases: 

• Discussing special cases such as right-angled triangles, where the Cosine Theorem reduces to the Pythagorean Theorem, can deepen understanding and illustrate specific scenarios.

3.5. Example problems: 

• Providing examples of problems where the Cosine Theorem is used to solve triangles or analyze geometric relationships can enhance comprehension and application skills.

3.6. Graphical Representations

• Visual representations, such as diagrams and graphs, can aid in understanding the geometric relationships described by the Cosine Theorem and its applications in solving triangles.

3.7. Real-world applications

• Exploring real-world contexts where the Cosine Theorem is applicable, such as navigation, surveying, and engineering, can demonstrate its relevance and practical utility.

4. Finding the Area of a Triangle:

4.1. Area of a Triangle:

• The area of a triangle is a measure of the region enclosed by its three sides.

• There are different methods to find the area of a triangle, depending on the given information.

4.2. Basic formula: 

• The most common formula to find the area of a triangle is:
   In this formula, the base and height of the triangle are perpendicular to each other.

4.3. Area using Side Lengths:

4.4. Area using Coordinates:

4.5. Area using Trigonometry:

4.6. Special cases:

4.7. Applications:

• Finding the area of a triangle is essential in various real-world applications, including geometry, engineering, architecture, and surveying. 

• It is used to calculate surface areas, volumes, and other geometric properties of objects and structures.

4.8. Practice Problems: 

• Solving practice problems involving different methods to find the area of triangles helps reinforce understanding and application of the concepts.

5. Radians, Arcs, and Circular Sectors:

5.1. Radian Measure: 

5.2. Arc Length:

5.3. Sector of a Circle:

• Definition: A sector of a circle is the region bounded by two radii and the arc between them.

5.4. Converting Between Radians and Degrees:

5.5 Applications:

   - Trigonometry: Radian measure simplifies calculations involving trigonometric functions such as sine, cosine, and tangent.

   - Calculus: Radian measure is preferred in calculus as it simplifies derivatives and integrals involving circular functions.

   - Physics: Radian measure is commonly used in physics to describe rotational motion, angular velocity, and angular displacement.

5.6. Practice Problems:

   - Solving practice problems involving radians, arc length, and circular sectors helps reinforce understanding and application of the concepts.

5.7. Trigonometric Identities:

6. Derivatives of trigonometric functions:

7. Integral of Sine and Cosine