We are reinforcing:

• conversion factors
• unit relationships
• radians/degrees
• dimensional analysis thinking

This is helpful in supporting unit logic before we get to the core trigonometry.

Quick Reminder

Degree ↔ Radian Conversion

$180^\circ = \pi \text{ radians}$

$\pi \text{ radians} = 180^\circ$

Conversion Factors

• Multiply by a fraction equal to 1 (multiply by a form of 1)
• Units should cancel

Example:

$90^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{2}$

We are reinforcing:

• trig ratios are NOT special-triangle-only
• trig works because of similarity
• trig is relationship-based

Core Reminder

$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$

$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$

$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$

Answer the Concept Question:

Why do trig ratios stay the same for similar triangles?

Main Idea:

“Trig describes position and rotation.”

Coordinate Point Relationship

cos = x

sin = y

tan = $\frac{\text{y}}{\text{x}}$

on the unit circle:

Key Anchor Values

Every point on the unit circle is 1 unit away from the origin

*Watch this video to help: https://youtu.be/1m9p9iubMLU?si=_AxmOfA0CjISpQBN

*Watch this video to understand more about sin and cos (starts at 2:13): https://youtu.be/1m9p9iubMLU?si=TzuXlZ3Dxd8cLL2J&t=133

Zero Degrees

At $0^\circ$ the point is: (1 , 0)

Therefore:

• cosine = 1
• sine = 0

90 Degrees / (π/2)

Point: (0 , 1)

Therefore:

• cosine = 0
• sine = 1

180 Degrees / π

Point: (−1 , 0)

Therefore:

• cosine = -1
• sine = 0

270 Degrees / (3π/2)

Point: (0 , -1)

Therefore:

• cosine = 0
• sine = - 1

360 Degrees / 2π

Point: (1 , 0)

Therefore:

• cosine = 1
• sine = 0

1 full rotation completed.

The sign on cos and sin values can be positive or negative depending on the quadrant you fall in when trying to “extrapolate” your coordinate point on the circle.

Summary of the main idea:

1. Degree to Radian Conversion (Easy)

Convert: $150^\circ$ to radians.

2. Radian to Degree Conversion (Easy)

Convert: $\frac{7\pi}{6}$ to degrees.

3. SOH-CAH-TOA Triangle (Easy)

Visual

Draw:

• right triangle (bottom right angle is 90 degrees(
• acute angle at bottom left labeled $\theta$
• opposite side of $\theta$ = 6
• hypotenuse = 10

Find:

• $\sin\theta$
• $\cos\theta$

4. Tangent Relationship (Easy-Medium)

Visual

Draw:

• right triangle
• adjacent = 8
• opposite = 15

Find:

• $\tan\theta$
• hypotenuse

5. Coordinate Point → Trig (Easy-Medium)

The point on the unit circle is:

$\left(\frac{\sqrt3}{2},\frac12\right)$

Find:

• cosine
• sine

6. Unit Circle Recognition (Medium)

Find: $\sin(0)$ and $\cos(2\pi)$

7. Hidden Rotation Question (Medium)

Find: $\sin(18\pi)$

8. Hidden Cosine Coordinate Question (Medium)

A point on the unit circle has coordinates: $\left(-\frac12,\frac{\sqrt3}{2}\right)$

What is $\cos(\theta)$ ?

9. Pythagorean + Trig Blend (Medium)

Visual

Draw:

• right triangle
• hypotenuse = 13
• one leg = 5

Find:

• missing side
• sine
• cosine
• tangent

10. SAT-Style Coordinate Geometry Trig (Medium-Hard)

*Watch this video:

Visual

Draw:

• coordinate plane
• point at (3,4)
• triangle from origin to point

If the segment from the origin to (3,4) forms angle $\theta$ with the x-axis, find:

• $\sin\theta$
• $\cos\theta$
• $\tan\theta$

11. Rotation & Quadrants (Medium-Hard)

If:

• cosine is negative
• sine is positive

which quadrant could the angle be in?

12. Special Triangle Recognition (Medium-Hard)

Visual

Draw:

30-60-90 triangle:

• short leg = 5

Find:

• long leg
• hypotenuse
• $\sin(30^\circ)$
• $\cos(30^\circ)$

13. SAT Trap — Tangent as Slope (Hard)

Visual

Draw:

• line through origin
• point (4,2)

Question:

If the line forms angle $\theta$ with the x-axis, what is $\tan(\theta)$ ?

14. Extrapolation Question (Hard but Conceptual)

Visual

Draw:

• coordinate plane
• point rotating around a circle
• start at (1,0)
• rotate counterclockwise $90^\circ$

Predict the new coordinate point.

What are:

• cosine
• sine

15. Real-World Navigation Question (Harder Conceptual)

Why might engineers use triangles and angles to estimate the location of satellites or GPS signals?

A. Because the Earth is flat, so triangles are the simplest way to measure straight distances.

B. Because using triangles and angles allows engineers to calculate unknown distances and positions based on known measurements, even on a curved surface like Earth.

C. Because satellites always send signals in the shape of triangles to Earth.

D. Because angles are easier to measure than distances, so no other information is needed.

1. Why might engineers use triangles and angles to estimate the location of satellites or GPS signals?

A. Because the Earth is flat, so triangles are the simplest way to measure straight distances.

B. Because using triangles and angles allows engineers to calculate unknown distances and positions based on known measurements, even on a curved surface like Earth.

C. Because satellites always send signals in the shape of triangles to Earth.

D. Because angles are easier to measure than distances, so no other information is needed.

2. Why are latitude and longitude useful in GPS systems?

A. They create a coordinate system that helps locate positions on Earth.

B. They divide Earth into equal triangles for measuring weather.

C. They only help pilots and astronauts navigate.

D. They measure the temperature of different locations on Earth.

3. In trigonometry, why do we use sine and cosine?

A. To memorize formulas without using coordinates.

B. To estimate positions and relationships using angles and triangles.

C. To avoid using graphs and coordinate planes.

D. To make all triangles into right triangles.

4. A GPS system receives signals from several satellites. Why is using multiple satellites more accurate than using only one?

A. More satellites create larger triangles, which automatically doubles accuracy.

B. Multiple satellites provide more angles and distance relationships to better estimate an unknown location.

C. One satellite can only measure height, not location.

D. Satellites cannot work unless there are at least four directly above the same city.

5. Why can the unit circle help describe movement and rotation?

A. Because every point on the circle represents a position based on an angle of rotation.

B. Because circles only work with degrees, not coordinates.

C. Because rotation only occurs inside circles.

D. Because all triangles fit perfectly inside the unit circle.

6. If a point rotates counterclockwise around the unit circle, what changes?

A. Only the x-coordinate changes.

B. Only the y-coordinate changes.

C. The x- and y-coordinates both change based on the angle of rotation.

D. The radius changes depending on the quadrant.

7. Why is trigonometry useful for navigation and mapping?

A. It allows us to estimate distances, directions, and positions using angles and coordinates.

B. It replaces the need for maps entirely.

C. It only works on flat surfaces.

D. It guarantees perfectly exact measurements in every situation.

8. What does it mean to “extrapolate” a point using trigonometry?

A. To randomly estimate where a point might be located.

B. To use known relationships, angles, and measurements to calculate an unknown position.

C. To convert every coordinate into radians.

D. To simplify a triangle into a square.

9. Why does the unit circle connect trigonometry to coordinate geometry?

A. Because every angle on the unit circle corresponds to an x-coordinate and y-coordinate.

B. Because coordinates only exist inside circles.

C. Because trigonometry ignores triangles and only uses circles.

D. Because the unit circle eliminates the need for graphs.

10. Engineers know the angle and distance between two points. Why can triangles help them find a third unknown point?

A. Triangles always contain enough information to estimate relationships between sides and angles.

B. Triangles automatically reveal exact GPS coordinates.

C. Every triangle contains equal sides.

D. Triangles remove the need for measurements.