Right Triangle Trigonometry Notes

Right Triangle Trigonometry

Introduction to Trigonometry

  • Trigonometry is the study of angle measurement (Greek origin).

  • It involves relationships among sides and angles of triangles.

  • This section focuses on solving geometry problems using trigonometry.

  • We will work exclusively with right triangles (triangles containing a 90-degree angle).

Triangle Labeling Conventions

  • Angles are represented by capital letters (A, B, C).

  • Side lengths are represented by lowercase letters (a, b, c).

  • Angle C is always the right angle.

  • Side c is always the hypotenuse (opposite the right angle).

  • Side a is opposite angle A, and side b is opposite angle B.

Trigonometric Ratios Definitions

  • Trigonometric ratios are formulas defined in terms of the lengths of sides of right triangles.

  • There are three primary trigonometric ratios: sine, cosine, and tangent.

Sine

  • Sine of angle A (sin A) is the ratio of the length of the side opposite angle A to the length of the hypotenuse.

  • sin A = \frac{opposite}{hypotenuse} = \frac{a}{c}

Cosine

  • Cosine of angle A (cos A) is the ratio of the length of the side adjacent to angle A to the length of the hypotenuse.

  • cos A = \frac{adjacent}{hypotenuse} = \frac{b}{c}

Tangent

  • Tangent of angle A (tan A) is the ratio of the length of the side opposite angle A to the length of the side adjacent to angle A.

  • tan A = \frac{opposite}{adjacent} = \frac{a}{b}

Mnemonic

  • SOHCAHTOA is a mnemonic device to remember the trigonometric ratios.

    • SOH: Sine = Opposite / Hypotenuse

    • CAH: Cosine = Adjacent / Hypotenuse

    • TOA: Tangent = Opposite / Adjacent

Example Problem: Finding Trig Ratios

  • Problem: Find the three trig ratios for angle B in a right triangle with side lengths 5, 12, and 13 (where 13 is the hypotenuse).

  • Steps:

    1. Identify the sides relative to angle B:

      • Opposite side = 5

      • Adjacent side = 12

      • Hypotenuse = 13 (calculate using the Pythagorean Theorem if not given: a^2 + b^2 = c^2)

    2. Calculate the trig ratios:

      • sin B = \frac{opposite}{hypotenuse} = \frac{5}{13}

      • cos B = \frac{adjacent}{hypotenuse} = \frac{12}{13}

      • tan B = \frac{opposite}{adjacent} = \frac{5}{12}

Application of Trig Ratios: Solving Problems

  • Trig ratios can be used to solve problems involving lengths of sides and angles in right triangles.

  • Example: Bracing a tree with a rope.

    • A rope is anchored 5 feet away from the trunk at an angle of 55 degrees with the ground.

    • Find the length of the rope (c) and the height (h) at which it should be attached to the tree.

Solution

  1. Diagram: Draw a right triangle representing the situation.

  2. Identify Trig Ratio: To find the height (h), use the tangent ratio.

    • tan(55^\circ) = \frac{h}{5}

    • h = 5 \cdot tan(55^\circ)

    • h \approx 7.1 \text{ feet}

  3. Alternative Method: Use cosine to find the length of the rope (c).

    • cos(55^\circ) = \frac{5}{c}

    • c = \frac{5}{cos(55^\circ)}

    • c \approx 8.7 \text{ feet}

  4. Alternative Approach: Recognize the third angle in the triangle is 35 degrees and use sine.

    • sin(35^\circ) = \frac{5}{c}

    • c = \frac{5}{sin(35^\circ)}

Example: Finding an Angle

  • Problem: A 20-foot ladder should be placed no steeper than 60 degrees. To safely climb onto a roof, the ladder should reach at least two feet over the roofline. Can this ladder be used to safely climb onto a 15-foot roof?

  • Solution:

    1. Diagram: Draw a diagram representing the ladder, roof, and ground.

    2. Identify Lengths: Ladder length = 20 feet. The ladder extends 2 feet over the roof, so the height from the ground to the ladder's base at the roofline is 15 feet + 2 feet = 17 feet. However, the triangle formed by the ground, the wall and the ladder has a height of 15 feet (opposite to angle A) and a hypotenuse of 18 feet (20 feet -2 feet). We're trying to see if using 17 feet as the opposite side of an angle is possible with a 20 foot hypotenuse at <= 60 degrees

    3. Apply Trig Ratio: Use sine to find the angle (A).

      • sin(A) = \frac{15}{18}

    4. Solve for the Angle: Use the inverse sine function.

      • A = sin^{-1}(\frac{15}{18})

      • A \approx 56.4^\circ

    5. Conclusion: Since 56.4 degrees is less than 60 degrees, the ladder can be safely used.