Discover the powerful world of trigonometric functions: sine, cosine, and tangent. Learn how these ratios unlock the secrets of right triangles and enable precise calculations.
The study of relationships between angles and sides in triangles
Trigonometry:
The branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles.
Key Concepts:
Ancient Origins:
Trigonometry was developed by ancient Greek mathematicians to solve problems in astronomy, navigation, and architecture.
Modern Applications:
Understanding the parts of a right triangle
Right Triangle with angle θ
Hypotenuse (c):
The longest side, opposite the right angle
Opposite Side (a):
The side opposite to the angle θ
Adjacent Side (b):
The side next to the angle θ (not the hypotenuse)
Sine, Cosine, and Tangent - the foundation of trigonometry
Definition:
sin θ = opposite / hypotenuse
sin θ = a / c
Memory Trick:
"Some Old Hippie Caught Another Hippie Tripping On Acid"
SOH = Sine = Opposite / Hypotenuse
Definition:
cos θ = adjacent / hypotenuse
cos θ = b / c
Memory Trick:
"Some Old Hippie Caught Another Hippie Tripping On Acid"
CAH = Cosine = Adjacent / Hypotenuse
Definition:
tan θ = opposite / adjacent
tan θ = a / b
Memory Trick:
"Some Old Hippie Caught Another Hippie Tripping On Acid"
TOA = Tangent = Opposite / Adjacent
The most important memory trick in trigonometry
SOH-CAH-TOA
"Some Old Hippie Caught Another Hippie Tripping On Acid"
sin θ = O/H
Sine = Opposite / Hypotenuse
cos θ = A/H
Cosine = Adjacent / Hypotenuse
tan θ = O/A
Tangent = Opposite / Adjacent
Calculating trigonometric ratios
Given: Right triangle with sides 3, 4, 5 and angle θ opposite the side of length 3.
Solution:
• Opposite = 3, Adjacent = 4, Hypotenuse = 5
• sin θ = 3/5 = 0.6
• cos θ = 4/5 = 0.8
• tan θ = 3/4 = 0.75
Given: Right triangle with hypotenuse = 10, angle = 30°, find the opposite side.
Solution:
• Use sin θ = opposite / hypotenuse
• sin 30° = opposite / 10
• 0.5 = opposite / 10
• opposite = 10 × 0.5 = 5
Apply your trigonometric knowledge
Given: Right triangle with sides 5, 12, 13. Find sin θ, cos θ, and tan θ where θ is opposite the side of length 5.
Solution:
• Opposite = 5, Adjacent = 12, Hypotenuse = 13
• sin θ = 5/13 ≈ 0.385
• cos θ = 12/13 ≈ 0.923
• tan θ = 5/12 ≈ 0.417
Given: Right triangle with adjacent side = 8, angle = 45°. Find the opposite side using tangent.
Solution:
• Use tan θ = opposite / adjacent
• tan 45° = opposite / 8
• 1 = opposite / 8
• opposite = 8