SOH CAH TOA is a mnemonic for remembering the definitions of the three primary trigonometric functions:
Sine (SOH): Sine of an angle = opposite side / hypotenuse
Cosine (CAH): Cosine of an angle = adjacent side / hypotenuse
Tangent (TOA): Tangent of an angle = opposite side / adjacent side
Understanding Triangle Components
In a right triangle:
Theta (θ): the angle of interest
Opposite Side: the side opposite to angle θ
Adjacent Side: the side next to angle θ
Hypotenuse: the longest side opposite the right angle
Pythagorean Theorem
The relationship between the sides of a right triangle:
Formula: A² + B² = C²
Where A and B are the legs of the triangle, and C is the hypotenuse.
The Six Trigonometric Functions
Sine (sin)
sin(θ) = Opposite / Hypotenuse
Cosine (cos)
cos(θ) = Adjacent / Hypotenuse
Tangent (tan)
tan(θ) = Opposite / Adjacent
Cosecant (csc)
csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite
Secant (sec)
sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent
Cotangent (cot)
cot(θ) = 1 / tan(θ) = Adjacent / Opposite
Examples of Right Triangle Problems
Example 1: Finding Sides and Trig Functions
Given: Opposite = 4, Adjacent = 3
Finding Hypotenuse:
3² + 4² = C² ⇒ 9 + 16 = 25 ⇒ C = 5
Trigonometric Functions:
sin(θ) = 4/5
cos(θ) = 3/5
tan(θ) = 4/3
csc(θ) = 5/4
sec(θ) = 5/3
cot(θ) = 3/4
Example 2: 8-15-17 Triangle
Given: A = 8, C = 17
Finding Missing Side B:
B = sqrt(17² - 8²) = sqrt(225) = 15
Trigonometric Functions:
sin(θ) = 15/17
cos(θ) = 8/17
tan(θ) = 15/8
csc(θ) = 17/15
sec(θ) = 17/8
cot(θ) = 8/15
Example 3: 15-20-25 Triangle
Given: Hypotenuse = 25, Opposite = 15
Finding Missing Side:
Missing side = sqrt(25² - 15²) = sqrt(400) = 20
Trigonometric Functions:
sin(θ) = 20/25 = 4/5
cos(θ) = 15/25 = 3/5
tan(θ) = 20/15 = 4/3
csc(θ) = 5/4
sec(θ) = 5/3
cot(θ) = 3/4
Finding Missing Sides Using Trig Functions
Example: Angle = 38°, Adjacent = 42
Use Tangent: tan(38°) = x / 42 ⇒ x = 42 * tan(38°) ⇒ x ≈ 32.8*
Finding Missing Angles Using Trig Functions
Example: Given opposite = 5, adjacent = 4
tan(θ) = 5/4 ⇒ θ = arctan(5/4) ⇒ θ ≈ 51.34°
Conclusion
Understanding and applying SOH CAH TOA is essential for solving right triangle problems.
Practice with various examples to master these concepts.
Additional Resources
For further learning, consider enrolling in the recommended trigonometry course on Udemy covering a comprehensive curriculum on angles, trigonometric functions, and more.