Mga Trigonometric Functions at Angles: Degrees at Radians

Trigonometry literally means "triangle measurement," originating from Greek words "trigonon" (triangle) and "metron" (measure). Though it began with triangles, modern trigonometry extends to circular functions and periodic phenomena.

In the Philippines, trigonometry has practical applications in engineering, architecture, navigation, and astronomy. Engineers designing bridges in Metro Manila use trigonometric calculations for precise angles and distances, while Philippine Airlines pilots rely on these principles for navigation.

The three basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions express ratios between the sides of a right triangle based on a given angle.

Fun Fact: The mathematical concepts we're learning today help solve real problems throughout the Philippines - from building structures that can withstand typhoons to navigating ships through our archipelago!

Mga Basic Trigonometric Functions

In a right triangle, we identify three sides: the hypotenuse (longest side), the opposite side (across from the angle), and the adjacent side (next to the angle). The three fundamental trigonometric functions represent ratios between these sides:

1. Sine (sin) = opposite ÷ hypotenuse
2. Cosine (cos) = adjacent ÷ hypotenuse
3. Tangent (tan) = opposite ÷ adjacent

Let's apply this to a real example: A 15-meter flagpole at Rizal Park creates a 30° angle of elevation when viewed from 26 meters away. We can calculate that sin 30° = 0.5, cos 30° ≈ 0.866, and tan 30° ≈ 0.577.

Some angles appear frequently in problems and should be memorized: 30°, 45°, and 60° (or π/6, π/4, and π/3 in radians).

Pro Tip: Memorizing the values for special angles (30°, 45°, 60°) will save you time during exams and make problem-solving much faster!

Special Angle Values

For 30° (π/6 radians):

• sin 30° = 1/2
• cos 30° = √3/2
• tan 30° = 1/√3

For 45° (π/4 radians):

• sin 45° = √2/2
• cos 45° = √2/2
• tan 45° = 1

For 60° (π/3 radians):

• sin 60° = √3/2
• cos 60° = 1/2
• tan 60° = √3

These special angle values form the foundation of trigonometric problem-solving. You'll use them repeatedly in various applications, from calculating heights and distances to solving complex physics problems.

Remember: These values aren't just random numbers to memorize - they're precise mathematical relationships derived from geometric principles. Understanding where they come from helps you remember them better!

Degrees at Radians: Mga Angle Measurements

In trigonometry, we measure angles in two main ways: degrees and radians. While degrees are more familiar in everyday life, radians are preferred in advanced mathematics.

The degree system divides a complete rotation into 360° - a convention from ancient Babylonians who used a base-60 number system. A radian, meanwhile, is a more natural unit directly related to circle properties. It's defined as the angle made by an arc with length equal to the radius.

Since a circle's circumference is 2πr, a complete rotation equals 2π radians. This gives us important equivalences:

• 360° = 2π radians
• 180° = π radians
• 90° = π/2 radians

To convert between the two systems, use these formulas:

• radians = degrees × $π/180$
• degrees = radians × $180/π$

Calculator Tip: Always check if your calculator is in the correct mode (degree or radian) before computing trigonometric functions!

Conversion Examples

Converting between degrees and radians is straightforward with practice. Here are some examples:

1. Converting 45° to radians: 45° × $π/180$ = π/4 radians

2. Converting π/3 radians to degrees: $π/3$ × $180/π$ = 60°

3. Converting 120° to radians: 120° × $π/180$ = 2π/3 radians

These conversions are essential when working with different formulas or when your calculator uses a different unit than your problem. Many scientific calculators show DEG or RAD indicators on screen to help you keep track of which mode you're using.

Practical Advice: When in doubt about which unit to use, radians are generally preferred for calculus and theoretical work, while degrees are often better for practical applications and visualization.

Unit Circle at Trigonometric Values

The unit circle is a circle with radius 1 centered at the origin (0,0) of the coordinate plane. It's a powerful visual tool for understanding trigonometric functions.

On the unit circle, the coordinates of any point are (cos θ, sin θ), where θ is the angle from the positive x-axis. This explains why sine and cosine always have values between -1 and 1.

Important angles on the unit circle and their coordinates include:

• 0° (0 radians): (1, 0)
• 30° (π/6 radians): $√3/2, 1/2$
• 45° (π/4 radians): $√2/2, √2/2$
• 90° (π/2 radians): (0, 1)
• 180° (π radians): $-1, 0$

The coordinate plane is divided into four quadrants, and the signs of trigonometric functions change in each:

• Quadrant I: All functions positive
• Quadrant II: Only sine positive
• Quadrant III: Only tangent positive
• Quadrant IV: Only cosine positive

Memory Aid: "All Students Take Calculus" can help you remember which functions are positive in each quadrant!

Finding Exact Values Using Quadrants

Understanding how trigonometric functions behave in different quadrants helps us find exact values for any angle. Let's look at an example:

To find sin 150° and cos 150°:

1. Identify that 150° is in Quadrant II, where sine is positive and cosine is negative
2. Recognize that 150° = 180° - 30°
3. Use reference angle: sin 150° = sin 30° = 1/2
4. Use reference angle with correct sign: cos 150° = -cos 30° = -√3/2

This approach works for any angle. First, identify the quadrant, then find the reference angle (the acute angle formed with the x-axis), and finally apply the correct sign based on the quadrant.

Visualization Tip: Drawing a quick sketch of the unit circle can help you determine the correct signs and values for any angle.

Mga Aplikasyon sa Real-World Problems

Trigonometry has numerous practical applications, especially relevant in the Philippine archipelago. Let's explore how trigonometric functions solve real problems.

For navigation and distance measurement, consider a boat traveling from Bataan to Corregidor Island at 35° North of East for 25 kilometers. To find how far east it traveled, we calculate: distance east = 25 × cos 35° = 20.48 kilometers.

In architecture and construction, an architect designing a roof in Baguio with a 30° angle and 12-meter horizontal span would calculate the height as: height = $12/2$ × tan 30° = 3.46 meters.

For astronomy, if a star is observed at a 65° elevation angle from the Manila Observatory (50 meters above sea level), the horizontal distance to where the star appears at horizon level would be: horizontal distance = 50/tan 65° ≈ 23.32 meters.

Problem-Solving Strategy: When tackling real-world problems, always draw a diagram first to visualize the situation and identify which trigonometric function applies!

Practice Problems at Review

Regular practice is essential for mastering trigonometric concepts. Here are some problems to strengthen your understanding:

Conversion Problems:

1. Convert from degrees to radians: a) 30° = π/6 radians b) 135° = 3π/4 radians

2. Convert from radians to degrees: a) π/6 = 30° b) 3π/4 = 135°

Trigonometric Function Problems: 3. Find exact values: a) sin 240° = -√3/2 b) cos 300° = 1/2

4. If sin θ = 3/5 and θ is in Quadrant II: cos θ = -4/5, tan θ = -3/4

Word Problems: 5. A helicopter flying 500 meters high observes a building at a 25° angle of depression. The horizontal distance is approximately 1073.2 meters.

Success Tip: Don't just check your answers - try to understand why you made mistakes. Regular practice and self-assessment are key to mastering trigonometry!