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Summary of Right Triangle Trigonometry and Trigonometric Functions

This summary covers the fundamental concepts of right triangle trigonometry, including the definitions of trigonometric functions, the use of the SOHCAHTOA mnemonic, special right triangles, and solving for unknown sides and angles. It also briefly outlines a trigonometry course curriculum.

I. Introduction to Right Triangle Trigonometry

• SOHCAHTOA: A mnemonic used to remember the basic trigonometric ratios:

  • Sin = Opposite / Hypotenuse
  • Cos = Adjacent / Hypotenuse
  • Tan = Opposite / Adjacent

• Sides of a Right Triangle (relative to an angle θ):

  • Opposite: The side directly across from the angle θ.
  • Adjacent: The side next to the angle θ (not the hypotenuse).
  • Hypotenuse: The longest side, opposite the right angle.

• Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c, $a^2 + b^2 = c^2$. This is used to find a missing side when two sides are known.

II. The Six Trigonometric Functions

The six trigonometric functions are defined based on the ratios of the sides of a right triangle:

• Sine (sin θ): Opposite / Hypotenuse
• Cosine (cos θ): Adjacent / Hypotenuse
• Tangent (tan θ): Opposite / Adjacent

The other three trigonometric functions are reciprocals of these:

• Cosecant (csc θ): 1 / sin θ = Hypotenuse / Opposite
• Secant (sec θ): 1 / cos θ = Hypotenuse / Adjacent
• Cotangent (cot θ): 1 / tan θ = Adjacent / Opposite

III. Special Right Triangles and Pythagorean Triples

• Pythagorean Triples: Sets of three integers that satisfy the Pythagorean theorem. Common examples include:
  • 3-4-5
  • 5-12-13
  • 8-15-17
  • 7-24-25
• Multiples of Triples: Any whole number multiple of a Pythagorean triple is also a valid set of sides for a right triangle (e.g., 6-8-10 is a multiple of 3-4-5).
• Less Common Triples: 9-40-41, 11-60-61. Recognizing these can help quickly find missing sides.

IV. Solving for Missing Sides and Trigonometric Functions

Example 1:

• Given a right triangle with sides 3 and 4, and angle θ.
• Find the missing side (hypotenuse): Using the Pythagorean theorem ($3^2 + 4^2 = c^2$), $9 + 16 = 25$, so $c = 5$.
• Calculate the six trig functions:
  • sin θ = Opposite/Hypotenuse = 4/5
  • cos θ = Adjacent/Hypotenuse = 3/5
  • tan θ = Opposite/Adjacent = 4/3
  • csc θ = 1/sin θ = 5/4
  • sec θ = 1/cos θ = 5/3
  • cot θ = 1/tan θ = 3/4

Example 2:

• Given a right triangle with sides 8 and 17, and angle θ.
• Find the missing side (leg): Using the Pythagorean theorem ($8^2 + b^2 = 17^2$), $64 + b^2 = 289$, so $b^2 = 225$, and $b = 15$.
• Calculate the six trig functions:
  • sin θ = Opposite/Hypotenuse = 15/17
  • cos θ = Adjacent/Hypotenuse = 8/17
  • tan θ = Opposite/Adjacent = 15/8
  • csc θ = 17/15
  • sec θ = 17/8
  • cot θ = 8/15

Example 3:

• Given a right triangle with hypotenuse 25 and one leg 15, and angle θ.
• Recognize the 3-4-5 pattern: This is a 5x multiple (3*5=15, 5*5=25). The missing side is 4*5 = 20.
• Calculate the six trig functions:
  • sin θ = Opposite/Hypotenuse = 20/25 = 4/5
  • cos θ = Adjacent/Hypotenuse = 15/25 = 3/5
  • tan θ = Opposite/Adjacent = 20/15 = 4/3
  • csc θ = 5/4
  • sec θ = 5/3
  • cot θ = 3/4

V. Solving for Unknown Sides Using Trigonometric Functions (with angles)

When an angle measure is given, trigonometric functions can be used to find unknown sides. Ensure the calculator is in the correct mode (degrees or radians).

• Problem Type 1: Finding a Leg

  • Given angle, adjacent side, find opposite side (x).
    • Use tangent: tan(angle) = Opposite/Adjacent.
    • $x = Adjacent \times \tan(\text{angle})$
    • Example: Angle = 38°, Adjacent = 42. $x = 42 \times \tan(38^\circ) \approx 32.8$.
  • Given angle, hypotenuse, find adjacent side (x).
    • Use cosine: cos(angle) = Adjacent/Hypotenuse.
    • $x = Hypotenuse \times \cos(\text{angle})$
    • Example: Angle = 54°, Hypotenuse = 26. $x = 26 \times \cos(54^\circ) \approx 15.28$.

• Problem Type 2: Finding the Hypotenuse

  • Given angle, opposite side, find hypotenuse (x).
    • Use sine: sin(angle) = Opposite/Hypotenuse.
    • $x = \text{Opposite} / \sin(\text{angle})$
    • Example: Angle = 32°, Opposite = 12. $x = 12 / \sin(32^\circ) \approx 22.64$.

VI. Solving for Unknown Angles Using Inverse Trigonometric Functions

When two sides are known, inverse trigonometric functions can be used to find the angle.

• Tangent: If tan θ = Opposite/Adjacent, then θ = arctan(Opposite/Adjacent).
  • Example: Opposite = 5, Adjacent = 4. θ = arctan(5/4) ≈ 51.34°.
• Cosine: If cos θ = Adjacent/Hypotenuse, then θ = arccos(Adjacent/Hypotenuse).
  • Example: Adjacent = 3, Hypotenuse = 7. θ = arccos(3/7) ≈ 64.62°.
• Sine: If sin θ = Opposite/Hypotenuse, then θ = arcsin(Opposite/Hypotenuse).
  • Example: Opposite = 5, Hypotenuse = 6. θ = arcsin(5/6) ≈ 56.44°.

VII. Trigonometry Course Curriculum Outline

The outline describes a comprehensive trigonometry course covering:

• Angles: Radians, degrees, conversions, coterminal angles, arc length, sector area, speed.
• Unit Circle: Six trig functions, reference angles.
• Right Triangle Trigonometry: SOHCAHTOA, special right triangles (30-60-90), solving triangles.
• Trigonometric Functions of Any Angle: Extending beyond right triangles.
• Graphs of Trig Functions: Sine, cosine, tangent, secant, cosecant.
• Inverse Trig Functions: Evaluation and graphing.
• Composition of Trig Functions: e.g., sin(arccos(x)).
• Applications: Problems involving two right triangles, bearings.
• Trigonometric Identities: Verifying identities.
• Sum/Difference, Double/Half Angle Formulas.
• Solving Trigonometric Equations.
• Future additions: Law of Sines, Law of Cosines, Polar Coordinates.