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Trigonometry - Trigonometric Functions

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    Study Guide Trigonometry – Trigonometric Functions 1. Angles 1.1 What Is an Angle? An angle is a measure of rotation. It tells us how much one line has turned away from another. Angles are measured in degrees ( °) . • One full rotation around a point equals 360° . An angle is formed by two rays: • The initial side is where the rotation starts. • The terminal side is where the rotation ends. 1.2 Positive and Negative Angles The direction of rotation matters. • Positive angles are formed by rotating counterclockwise . • Negative angles are formed by rotating clockwise . So, whether an angle is positive or negative depends only on the direction of rotation , not its size. 1.3 Standard Position Figure 1 ( a) A positive angle and ( b) a negative angle.

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    Study Guide An angle is said to be in standard position when: • Its initial side lies along the positive x - axis . • The vertex of the angle is at the origin ( 0, 0) . Placing angles in standard position helps us describe and compare them more easily. 1.4 Quadrantal Angles Figure 2 Types of Angles

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    Study Guide Some angles have their terminal side exactly on one of the coordinate axes . These are called quadrantal angles . Examples include: • 0° • 90° • 180° • 270° • Any angle whose terminal side lies directly on the x - axis or y - axis 1.5 Angles in the Four Quadrants If an angle in standard position is not quadrantal , then its terminal side lies in one of the four quadrants of the coordinate plane. Here’s how angles are classified based on where the terminal side ends up: • First Quadrant : Terminal side is between the positive x - axis and positive y - axis • ( Example: 40°) • Second Quadrant : Terminal side is between the positive y - axis and negative x - axis • ( Example: −260°) • Third Quadrant : Terminal side is between the negative x - axis and negative y - axis • ( Example: 210°) • Fourth Quadrant : Terminal side is between the negative y - axis and positive x - axis • ( Example: −50°)

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    Study Guide 1.6 Understanding Angles in Standard Position When an angle is drawn in standard position , its vertex is at the origin ( 0, 0) and its initial side lies along the positive x - axis . From there, the angle opens either counterclockwise ( positive angles) or clockwise ( negative angles) . Where the terminal side of the angle ends tells us which quadrant the angle is in. Example 1: Identifying the Quadrant of an Angle

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    Study Guide Figure 3 Angles coterminal with 30° 1.7 What Does Coterminal Mean? Two angles are called coterminal if: • They are in standard position , and • They share the same terminal side . In other words, they may look different, but they end in exactly the same place .

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    Study Guide 1.8 Coterminal Angles and Full Rotations A full rotation around a circle is 360° . So, you can create coterminal angles by adding or subtracting 360° . General Rule: All angles that are coterminal with an angle ( d° ) can be written as: where ( n ) is any integer ( positive, negative, or zero). Example: Angles Coterminal with 30° All the angles below share the same terminal side as 30° , even though they look very different: Even though these angles involve multiple rotations, they all end in the same position . 1.9 What Does “Coterminal” Mean? Two angles are called coterminal angles if they: • Start at the same initial side • End at the same terminal side This happens when the angles differ by a whole number multiple of 360° . In general: where n is any integer ( positive, negative, or zero). Example 2: Are 200° and 940° Coterminal? To check if two angles are coterminal, subtract one from the other and see if the result is a multiple of 360°. If 940° and 200° were coterminal:

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    Study Guide Subtract 200° from both sides: Since 740 is not a multiple of 360 , there is no whole number value of n . Conclusion: 200° and 940° are not coterminal angles. Example 3: Name Five Angles Coterminal with −70° To find coterminal angles, add or subtract multiples of 360°. Starting with −70°: • −70° + ( 1 × 360°) = 290° • −70° + ( 2 × 360°) = 650° • −70° + ( 3 × 360°) = 1010° • −70° + ( − 1 × 360°) = − 430° • −70° + ( − 2 × 360°) = − 790° All of these angles share the same terminal side as −70°, so they are coterminal. 1.10 Degrees, Minutes, and Seconds Angle measurements are not always whole degrees. Fractional angles can be written in two main ways: • As decimal degrees , such as 34.25° • Using degrees, minutes, and seconds ( DMS) 1.11 Understanding Minutes and Seconds The degree is divided into smaller units: • 1 degree ( °) = 60 minutes ( ′) • 1 minute ( ′) = 60 seconds ( ″)

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    Study Guide In symbols: • 1° = 60′ • 1′ = 60″ 1.12 Converting Between Forms Example 4: Write 34°15′ Using Decimal Degrees Convert minutes to degrees by dividing by 60. Example 5: Write 12°18′44″ Using Decimal Degrees Convert minutes and seconds step by step. Example 6: Write 81.293° Using Degrees, Minutes, and Seconds Start with the whole number part: Convert the decimal part to minutes: So: Convert the decimal minutes to seconds:

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    Study Guide Final Answer: Summary • Coterminal angles differ by multiples of 360° • Minutes and seconds are smaller units of degrees • You can convert between decimal degrees and DMS by multiplying or dividing by 60 • These conversions are essential for accuracy in geometry and trigonometry 2. Functions of Acute Angles 2.1 Why Similar Triangles Matter Trigonometry is built on the idea of similar triangles , first studied by Euclid. Two triangles are similar if: • Their corresponding angles are equal, and • Their corresponding sides are in the same ratio. Because all right triangles have a 90° angle , any two right triangles that share another equal angle must be similar. This means the ratios of their corresponding sides are the same , even if the triangles are different sizes. This idea leads directly to trigonometric ratios . 2.2 Naming Angles Angles are usually represented using lowercase Greek letters . The most common ones are: • α ( alpha) • θ ( theta) The letter used doesn’t change the math — it’s just a name for the angle.

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    Study Guide 2.3 Measuring Angles Angles can be measured in degrees or radians . The relationship between them is: • 180° = π radians • 1° = π / 180 radians • 1 radian = 180° / π You’ll often switch between these units in trigonometry, so it’s important to know these conversions. 2.4 Triangles and the Circle To define trigonometric ratios, we often use a right triangle placed inside a circle with equation: Figure 1 Reference triangles. In this triangle: • r is the hypotenuse • x is the side adjacent to the angle • y is the side opposite the angle

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