Lesson Description
This lesson applies trigonometric ratios—sine, cosine, and tangent—to solve contextual word problems involving right triangles. Emphasis is placed on modeling, equation setup, calculator usage, and interpretation of results.
Learning Objectives
- Identify appropriate trig ratios based on known information.
- Set up and solve equations using sine, cosine, and tangent.
- Interpret solutions within real-world contexts.
Topics Covered
- \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
- \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
- Angle of elevation and angle of depression
- Calculator use and rounding conventions
Key Terms
- Sine (sin): Ratio of opposite side to hypotenuse.
- Cosine (cos): Ratio of adjacent side to hypotenuse.
- Tangent (tan): Ratio of opposite side to adjacent side.
- Angle of Elevation: Angle measured upward from horizontal.
- Angle of Depression: Angle measured downward from horizontal.
Lesson Examples
Example 1: Finding Height
A ladder leans against a wall forming a 65° angle with the ground. The ladder is 10 ft long. How high up the wall does it reach?
\[ \sin(65^\circ) = \frac{h}{10} \]
\[ h = 10\sin(65^\circ) \approx 9.06 \text{ ft} \]
Example 2: Finding Distance
A tree casts a shadow 15 ft long. The angle of elevation of the sun is 40°. Find the height of the tree.
\[ \tan(40^\circ) = \frac{h}{15} \]
\[ h = 15\tan(40^\circ) \approx 12.59 \text{ ft} \]
Practice Problems
- A flagpole casts a 20 ft shadow. The angle of elevation is 35°. Find its height.
- A plane climbs at 12° for 3 miles. How high is it above the ground?
- From a 50 ft building, the angle of depression to a car is 25°. How far is the car from the building?
Tips & Common Mistakes
- Check that your calculator is in degree mode.
- Label triangles clearly before choosing a ratio.
- Round only at the final step unless instructed otherwise.
Summary
Trig ratios allow us to convert angle measures into measurable distances. Proper setup, ratio selection, and interpretation are essential for accurate modeling.
Challenge Problems
- A drone rises at a 28° angle. After traveling 500 ft, how high is it?
- A 120 ft tower is observed from 200 ft away. What is the angle of elevation?
- Explain why sine is used instead of cosine in Example 1.