The Unit Circle: Your Trigonometry Secret Weapon 🚀

Hey there, future math master! Let's dive into the unit circle – your best friend for tackling those tricky trig questions on the SAT. Think of it as a visual cheat sheet that unlocks a world of understanding. This guide is designed to be your go-to resource the night before the exam, so let's make every minute count!

Unit Circle Basics

What is the Unit Circle?

• A circle with a radius of 1 unit, centered at the origin (0, 0). It’s your visual playground for angles and trig functions.
• Angles are measured in radians (full circle = 2π) or degrees (full circle = 360°).
• We always start measuring angles counterclockwise from the positive x-axis.
• The coordinates of any point on the circle are given by (cos(θ), sin(θ)), where θ is the angle.
• The unit circle's equation, $$x^2 + y^2 = 1$$, comes straight from the Pythagorean theorem. 💡

Angle-Coordinate Relationship

• The angle's initial side is always on the positive x-axis.
• The terminal side is where the angle measurement stops.
• The x-coordinate of the point where the terminal side intersects the circle is equal to cos(θ).
• The y-coordinate of the point where the terminal side intersects the circle is equal to sin(θ).
• For example, a 90° (or π/2 radians) angle puts you at the point (0, 1).

Sine, Cosine, and Tangent Values

Common Angle Values

• You absolutely need to memorize the sine, cosine, and tangent values for these key angles: 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°.
• Their radian equivalents are: 0, π/6, π/4, π/3, π/2, π, 3π/2, 2π.
• Remember those special right triangles? (30-60-90 and 45-45-90) They're your secret to figuring out these values!
• For instance: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3.

Trigonometric Ratios

• For an angle θ in the first quadrant:
  • $$sin(θ) = \frac{opposite}{hypotenuse} = y-coordinate$$
  • $$cos(θ) = \frac{adjacent}{hypotenuse} = x-coordinate$$
  • $$tan(θ) = \frac{opposite}{adjacent} = \frac{y-coordinate}{x-coordinate}$$

- - Example: In the second quadrant, sine is positive, but cosine and tangent are negative.

Trigonometric Ratios in Right Triangles

Applying Ratios to Find Side Lengths

• Use trig ratios to find unknown sides in right triangles.

  • Sine ratio: $$sin(θ) = \frac{opposite}{hypotenuse}$$
  • Cosine ratio: $$cos(θ) = \frac{adjacent}{hypotenuse}$$
  • Tangent ratio: $$tan(θ) = \frac{opposite}{adjacent}$$

• Example: In a right triangle with a 30° angle and a hypotenuse of 10, the opposite side is 10 * sin(30°) = 5. ### Solving for Angles and Sides

• If you know one angle and one side, you can find the missing sides using trig ratios.

• Inverse trig functions (arcsin, arccos, arctan) help you find angles when you know side ratios.

• Example: If opposite/hypotenuse = 0.5, then the angle θ = arcsin(0.5) ≈ 30°.

• Don't forget the Pythagorean theorem, $$a^2 + b^2 = c^2$$, it's your backup for side calculations!

Final Exam Focus

High-Priority Topics

• Unit Circle Basics: Understand its structure, angles, and coordinates.
• Common Angle Values: Memorize the key values for sine, cosine, and tangent.
• Trigonometric Ratios: Master SOHCAHTOA and ASTC.
• Right Triangle Applications: Be ready to solve for sides and angles.

Common Question Types

• Multiple Choice: Expect questions testing your knowledge of unit circle coordinates, trig ratios, and special angles.
• Free Response: These often involve applying trig ratios in right triangles, sometimes with a real-world context.

Last-Minute Tips

• Time Management: Don't spend too long on any one question. If you're stuck, move on and come back later.
• Common Pitfalls: Watch out for sign errors in different quadrants. Double-check your calculations.
• Strategies: Visualize the unit circle for every trig question. It's your visual guide to success!

Practice Question

Practice Questions

Multiple Choice Questions

1. What is the value of $$cos(\frac{5\pi}{6})$$? (A) $$\frac{\sqrt{3}}{2}$$ (B) $$-\frac{\sqrt{3}}{2}$$ (C) $$\frac{1}{2}$$ (D) $$-\frac{1}{2}$$

2. In a right triangle, if $$sin(θ) = \frac{3}{5}$$, what is the value of $$cos(θ)$$? (A) $$\frac{4}{5}$$ (B) $$\frac{3}{4}$$ (C) $$\frac{5}{4}$$ (D) $$\frac{4}{3}$$

3. The point (\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}) lies on the unit circle. What is the angle in radians? (A) $$\frac{\pi}{4}$$ (B) $$\frac{3\pi}{4}$$ (C) $$\frac{5\pi}{4}$$ (D) $$\frac{7\pi}{4}$$

Free Response Question

A surveyor is standing 50 meters from the base of a building. The angle of elevation to the top of the building is 60 degrees.

(a) Draw a diagram representing the situation. (1 point)

(b) Calculate the height of the building. (2 points)

(c) If the surveyor walks 20 meters closer to the building, what is the new angle of elevation? (2 points)

(Scoring Breakdown)

(a) Diagram (1 point)

• A right triangle with the base as 50 meters, the angle of elevation as 60 degrees, and the height of the building as the unknown side. 1 point for a correct diagram.

(b) Height of the building (2 points)

• $$tan(60°) = \frac{height}{50}$$
• $$height = 50 * tan(60°) = 50 * \sqrt{3} ≈ 86.6$$ meters. 1 point for the correct setup and 1 point for the correct answer.

(c) New angle of elevation (2 points)

• New distance = 50 - 20 = 30 meters.
• $$tan(θ) = \frac{86.6}{30}$$
• $$θ = arctan(2.887) ≈ 70.9°$$. 1 point for the correct setup and 1 point for the correct answer.

Remember, you've got this! Go into the exam with confidence, and let your knowledge of the unit circle shine! ✨