#AP Pre-Calculus: Sine, Cosine, and Tangent - Your Last-Minute Guide 🚀

Hey there! Let's make sure you're feeling super confident about sine, cosine, and tangent for your AP Pre-Calculus exam tomorrow. This guide is designed to be quick, clear, and effective, just what you need right now!

#📐 Angle Basics: Setting the Stage

Let's start with the fundamentals:

• Standard Position: An angle is in standard position when its vertex is at the origin and its initial ray is along the positive x-axis.

  

  Initial and terminal rays

  
  
  • The initial ray is like the starting line, and the terminal ray is where the angle ends.

• Positive Angles: ⬆️ Measured by rotating counterclockwise from the positive x-axis.

• Negative Angles: ⬇️ Measured by rotating clockwise from the positive x-axis.

• Coterminal Angles: Angles that share the same terminal ray. They differ by multiples of 360° (or 2π radians). Think of it like laps around a track – you end up at the same place, but you've traveled different distances. 🔄

#🗣️ Trigonometry Talk: Degrees vs. Radians

#▶️ Radians: The Cool Kids of Angle Measurement

• Definition: Radians measure angles based on the arc length they subtend on a circle. It's the ratio of the arc length to the radius. 📏

  

  Radian Measurement on a Unit Circle

• Unit Circle: When the radius is 1, the radian measure equals the arc length. Super convenient! 💡

• Why Radians? They're more natural in math and physics because they're based on the geometry of the circle, not an arbitrary division like degrees. 🤓

#1️⃣ Sine (sin): The Vertical Vibe

• Definition: The sine of an angle is the ratio of the y-coordinate of a point on the unit circle to the radius. sin(θ) = y/r
• Unit Circle: On the unit circle (radius = 1), sin(θ) is simply the y-coordinate. ↕️
• Periodicity: The sine function repeats every 2π radians. It's like a wave that keeps going up and down. 🌊
• Range: The values of sine are always between -1 and 1. [-1, 1]

#2️⃣ Cosine (cos): The Horizontal Hero

• Definition: The cosine of an angle is the ratio of the x-coordinate of a point on the unit circle to the radius. cos(θ) = x/r
• Unit Circle: On the unit circle, cos(θ) is simply the x-coordinate. ↔️
• Periodicity: Like sine, cosine repeats every 2π radians. 🔄
• Range: The values of cosine are also between -1 and 1. [-1, 1]

*Cosine and Sine on a Unit Circle*

#3️⃣ Tangent (tan): The Slope Star

• Definition: The tangent of an angle is the slope of the terminal ray. It's the ratio of the y-coordinate to the x-coordinate. tan(θ) = y/x
• Alternative Definition: Tangent can also be defined as sin(θ) / cos(θ). tan(θ) = sin(θ) / cos(θ) ↗️
• Periodicity: The tangent function repeats every π radians. It's a bit faster than sine and cosine. 💨
• Range: Tangent has no defined range. It can be any real number. It goes all the way up and all the way down. 🎢

#Final Exam Focus

#High-Priority Topics:

• Unit Circle: Know your coordinates for key angles like 0, π/6, π/4, π/3, π/2, and their multiples. 🎯
• Radian-Degree Conversions: Be quick and accurate in converting between radians and degrees. 🔄
• Trigonometric Identities: Understand the basic relationships between sin, cos, and tan. 🔗
• Graphs of Trig Functions: Recognize the shapes and key features of sine, cosine, and tangent graphs. 📈

#Common Question Types:

• Multiple Choice: Expect questions that test your understanding of definitions and basic calculations. 🧐
• Free Response: Be prepared to apply your knowledge to solve multi-step problems involving trigonometric functions. 📝

#Last-Minute Tips:

• Time Management: Don't get stuck on a single question. Move on and come back if you have time. ⏱️
• Common Pitfalls: Watch out for sign errors and undefined values. 👀
• Strategies: Draw diagrams to visualize problems. Use your calculator wisely. 🧮
• Stay Calm: You've got this! Take deep breaths and trust your preparation. 🧘

#Practice Questions

Practice Question

#Multiple Choice Questions

1. What is the radian measure of an angle that measures 135 degrees? (A) 3π/4 (B) 5π/6 (C) 7π/12 (D) 2π/3

2. If sin(θ) = 0.6 and θ is in the second quadrant, what is the value of cos(θ)? (A) 0.8 (B) -0.8 (C) 0.4 (D) -0.4

3. What is the period of the function f(x) = tan(2x)? (A) π (B) 2π (C) π/2 (D) π/4

#Free Response Question

Consider the function f(x) = 2sin(x)cos(x)

(a) Rewrite the function using a double-angle identity. (1 point)

(b) Determine the amplitude and period of the function. (2 points)

(c) Sketch the graph of the function over the interval [0, 2π]. (3 points)

(d) Find all values of x in the interval [0, 2π] where f(x) = 1. (2 points)

Scoring Breakdown:

(a) 1 point for correctly using the double-angle identity: f(x) = sin(2x)

(b) 1 point for stating the amplitude is 1 1 point for stating the period is π

(c) 1 point for correct shape of the sine wave 1 point for correct amplitude 1 point for correct period

(d) 1 point for setting up the equation sin(2x) = 1 1 point for finding the solutions x = π/4 and x = 5π/4

Alright, you've got this! Go ace that exam! 💪