Introduction to Trigonometry

Introduction to Trigonometry

• Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles.
• The word "trigonometry" comes from the Greek words "tri" meaning three, "gon" meaning angles, and "metry" meaning measurement.
• Trigonometry involves the use of trigonometric ratios, which are the ratios of the sides of a right-angled triangle.
• Trigonometric ratios are used to solve problems in various fields such as physics, engineering, navigation, and mathematics.

Trigonometric Ratios

• Sine (sin): The ratio of the opposite side to the hypotenuse of a right-angled triangle.
• Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right-angled triangle.
• Tangent (tan): The ratio of the opposite side to the adjacent side of a right-angled triangle.
• Cotangent (cot): The ratio of the adjacent side to the opposite side of a right-angled triangle.
• Secant (sec): The ratio of the hypotenuse to the opposite side of a right-angled triangle.
• Cosecant (cosec): The ratio of the hypotenuse to the adjacent side of a right-angled triangle.

Triangles

BPT Theorem

• Basic Proportionality Theorem (BPT): If a straight line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• The theorem states that if DE is parallel to BC, then:
  • DE / BC = AD / AB
  • DE / AC = BD / BC
• BPT is used to prove various theorems in geometry and to solve problems related to triangles.

Similarity of Triangles

• Similar Triangles: Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional.
• Similarity Criteria:
  • AAA (Angle-Angle-Angle): If two triangles have three equal pairs of angles, they are similar.
  • AA (Angle-Angle): If two triangles have two equal pairs of angles and the sides including them are proportional, they are similar.
  • SSS (Side-Side-Side): If two triangles have three pairs of proportional sides, they are similar.
  • SAS (Side-Angle-Side): If two triangles have two pairs of proportional sides and the included angle is equal, they are similar.
• Properties of Similar Triangles:
  • Corresponding angles are equal.
  • Corresponding sides are proportional.
  • The ratio of the areas of the two triangles is equal to the square of the ratio of their corresponding sides.

Introduction to Trigonometry

• Trigonometry deals with relationships between sides and angles of triangles.
• The word "trigonometry" comes from Greek words "tri" meaning three, "gon" meaning angles, and "metry" meaning measurement.

Trigonometric Ratios

• Sine (sin) is the ratio of the opposite side to the hypotenuse of a right-angled triangle.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse of a right-angled triangle.
• Tangent (tan) is the ratio of the opposite side to the adjacent side of a right-angled triangle.
• Cotangent (cot) is the ratio of the adjacent side to the opposite side of a right-angled triangle.
• Secant (sec) is the ratio of the hypotenuse to the opposite side of a right-angled triangle.
• Cosecant (cosec) is the ratio of the hypotenuse to the adjacent side of a right-angled triangle.

Triangles

BPT Theorem

• Basic Proportionality Theorem (BPT) states that if a straight line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
• If DE is parallel to BC, then DE / BC = AD / AB and DE / AC = BD / BC.

Similarity of Triangles

• Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
• Similarity Criteria:
  • AAA (Angle-Angle-Angle): Two triangles are similar if they have three equal pairs of angles.
  • AA (Angle-Angle): Two triangles are similar if they have two equal pairs of angles and the sides including them are proportional.
  • SSS (Side-Side-Side): Two triangles are similar if they have three pairs of proportional sides.
  • SAS (Side-Angle-Side): Two triangles are similar if they have two pairs of proportional sides and the included angle is equal.
• Properties of Similar Triangles:
  • Corresponding angles are equal.
  • Corresponding sides are proportional.
  • The ratio of the areas of the two triangles is equal to the square of the ratio of their corresponding sides.

Introduction to Trigonometry

• Trigonometry deals with the relationships between the sides and angles of triangles.
• It involves trigonometric ratios, which are the ratios of the sides of a right-angled triangle.

Trigonometric Identities

• The Pythagorean Identity is sin^2(A) + cos^2(A) = 1.
• The Sum and Difference Identities are: • sin(A + B) = sin(A)cos(B) + cos(A)sin(B) • sin(A - B) = sin(A)cos(B) - cos(A)sin(B) • cos(A + B) = cos(A)cos(B) - sin(A)sin(B) • cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
• The Product Identities are: • sin(A)cos(B) = (1/2)[sin(A + B) + sin(A - B)] • cos(A)sin(B) = (1/2)[sin(A + B) - sin(A - B)]

Sine, Cosine, and Tangent

• 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.
• The Reciprocal Identities are: • csc(A) = 1/sin(A) • sec(A) = 1/cos(A) • cot(A) = 1/tan(A)

Triangle Properties

• The sum of the interior angles of a triangle is 180° (Angle Sum Property).
• The exterior angle of a triangle is equal to the sum of the two interior angles (Exterior Angle Property).
• The sum of any two sides of a triangle is greater than the third side (Triangle Inequality).

Triangles - BPT Theorem

• The BPT Theorem states that if a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same ratio.
• The Converse of BPT states that if a line divides the two sides of a triangle in the same ratio, then it is parallel to the third side.

Similarities of Triangles

• Similar triangles have the same shape and size but not necessarily the same orientation.
• The Similarity Criteria are: • AA (Angle-Angle): Two triangles with two pairs of congruent angles are similar. • SSS (Side-Side-Side): Two triangles with three pairs of congruent sides are similar. • SAS (Side-Angle-Side): Two triangles with two pairs of congruent sides and the included angle congruent are similar.