quartiles

Trigonometric Functions of Special Angles

Special Right Triangles

• 30-60-90 Triangles:
  • Hypotenuse: 2 times the length of the shorter leg
  • Longer leg: √3 times the length of the shorter leg
  • Trigonometric values:
    • sin(30°) = 1/2
    • cos(30°) = √3/2
    • tan(30°) = 1/√3
• 45-45-90 Triangles:
  • Legs are equal in length
  • Hypotenuse: √2 times the length of a leg
  • Trigonometric values:
    • sin(45°) = 1/√2
    • cos(45°) = 1/√2
    • tan(45°) = 1

Reference Angles

• An angle in the unit circle that has the same trigonometric values as the original angle
• Found by subtracting the angle from the closest multiple of 360° or adding the angle to the closest multiple of 360° *Used to simplify trigonometric expressions and find exact values

Unit Circle

• A circle with a radius of 1 unit, used to define trigonometric functions
• Centered at the origin (0, 0) of a coordinate plane
• Angles measured counterclockwise from the positive x-axis
• Used to visualize and calculate trigonometric functions

Exact Values

• sin(0°) = 0
• cos(0°) = 1
• tan(0°) = 0
• sin(30°) = 1/2
• cos(30°) = √3/2
• tan(30°) = 1/√3
• sin(45°) = 1/√2
• cos(45°) = 1/√2
• tan(45°) = 1
• sin(60°) = √3/2
• cos(60°) = 1/2
• tan(60°) = √3

Trigonometric Functions of Special Angles

Special Right Triangles

• 30-60-90 Triangles: Hypotenuse is 2 times the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg.
• 30-60-90 Triangles: Trigonometric values are sin(30°) = 1/2, cos(30°) = √3/2, and tan(30°) = 1/√3.
• 45-45-90 Triangles: Legs are equal in length, and the hypotenuse is √2 times the length of a leg.
• 45-45-90 Triangles: Trigonometric values are sin(45°) = 1/√2, cos(45°) = 1/√2, and tan(45°) = 1.

Reference Angles

• A reference angle is an angle in the unit circle with the same trigonometric values as the original angle.
• To find a reference angle, subtract the angle from the closest multiple of 360° or add the angle to the closest multiple of 360°.
• Reference angles are used to simplify trigonometric expressions and find exact values.

Unit Circle

• The unit circle is a circle with a radius of 1 unit, used to define trigonometric functions.
• The unit circle is centered at the origin (0, 0) of a coordinate plane.
• Angles are measured counterclockwise from the positive x-axis.
• The unit circle is used to visualize and calculate trigonometric functions.

Exact Values

• sin(0°) = 0, cos(0°) = 1, and tan(0°) = 0.
• sin(30°) = 1/2, cos(30°) = √3/2, and tan(30°) = 1/√3.
• sin(45°) = 1/√2, cos(45°) = 1/√2, and tan(45°) = 1.
• sin(60°) = √3/2, cos(60°) = 1/2, and tan(60°) = √3.

Trigonometric Functions

• Sine (sin): Ratio of opposite side to hypotenuse in a right-angled triangle
• Formula: sin(A) = opposite side / hypotenuse
• Range: -1 ≤ sin(A) ≤ 1
• Period: 2π (sin(A + 2π) = sin(A))
• Identities:
  • sin(A) = cos(π/2 - A)
  • sin(-A) = -sin(A)

Cosine (cos)

• Definition: Ratio of adjacent side to hypotenuse in a right-angled triangle
• Formula: cos(A) = adjacent side / hypotenuse
• Range: -1 ≤ cos(A) ≤ 1
• Period: 2π (cos(A + 2π) = cos(A))
• Identities:
  • cos(A) = sin(π/2 - A)
  • cos(-A) = cos(A)

Tangent (tan)

• Definition: Ratio of sine to cosine of an angle
• Formula: tan(A) = sin(A) / cos(A)
• Range: All real numbers (-∞ to ∞)
• Period: π (tan(A + π) = tan(A))
• Identities:
  • tan(A) = -tan(-A)
  • tan(A) = 1 / tan(π/2 - A)

Relationships between sine, cosine, and tangent

• Pythagorean Identity: sin^2(A) + cos^2(A) = 1
• Tan in terms of sin and cos: tan(A) = sin(A) / cos(A)
• sin and cos in terms of tan:
  • sin(A) = tan(A) / sqrt(1 + tan^2(A))
  • cos(A) = 1 / sqrt(1 + tan^2(A))

Learn about trigonometric functions of special angles, including 30-60-90 and 45-45-90 triangles, and their reference angles. Understand the relationships between sides and angles in these triangles.