Introduction to Trigonometry
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What is the main focus of the branch of mathematics called trigonometry?

  • calculating the area of polygons
  • dealing with the relationships between the sides and angles of triangles (correct)
  • studying the relationships between the sides and angles of quadrilaterals
  • measuring the perimeter of circles

    • What is the ratio of the opposite side to the adjacent side of a right-angled triangle?

  • tangent (tan) (correct)
  • cosine (cos)
  • sine (sin)
  • cotangent (cot)

    • If DE is parallel to BC in a triangle, what is the ratio of DE to BC according to the Basic Proportionality Theorem (BPT)?

  • AD / AC
  • DE / BC = BC / AB
  • AD / AB (correct)
  • BD / BC

    • What is the condition for two triangles to be similar according to the AAA (Angle-Angle-Angle) criterion?

    <p>three equal pairs of angles</p> Signup and view all the answers

    What is the term for the ratio of the hypotenuse to the opposite side of a right-angled triangle?

    <p>secant (sec)</p> Signup and view all the answers

    If two triangles have two equal pairs of angles and the sides including them are proportional, what criterion is used to determine their similarity?

    <p>AA (Angle-Angle)</p> Signup and view all the answers

    What is the ratio of the hypotenuse to the adjacent side of a right-angled triangle?

    <p>cosecant (cosec)</p> Signup and view all the answers

    What is the term for the ratio of the adjacent side to the opposite side of a right-angled triangle?

    <p>cotangent (cot)</p> Signup and view all the answers

    What is the Pythagorean Identity in trigonometry?

    <p>sin^2(A) + cos^2(A) = 1</p> Signup and view all the answers

    The sum of the interior angles of a triangle is always 90°.

    <p>False</p> Signup and view all the answers

    What is the value of sin(A)cos(B) + cos(A)sin(B) in trigonometry?

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

    The reciprocal of sine is called _______________.

    <p>cosecant (csc)</p> Signup and view all the answers

    Match the following trigonometric ratios with their definitions:

    <p>sin(A) = opposite side / hypotenuse cos(A) = adjacent side / hypotenuse tan(A) = opposite side / adjacent side</p> Signup and view all the answers

    What is the condition for two triangles to be similar according to the AA criterion?

    <p>Two pairs of congruent angles</p> Signup and view all the answers

    If a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same ratio.

    <p>True</p> Signup and view all the answers

    What is the term for the study of relationships between the sides and angles of triangles?

    <p>trigonometry</p> Signup and view all the answers

    Study Notes

    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.

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