Algebra 3 - Trigonometry Flashcards
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Questions and Answers

What is the term for 'Sine'?

Sine

What is the term for 'Cosine'?

Cosine

What is the term for 'Tangent'?

Tangent

What is the value of 'Sine 30'?

<p>0.5</p> Signup and view all the answers

What is the value of 'Sine 60'?

<p>√3/2</p> Signup and view all the answers

What is the value of 'Sine 45'?

<p>√2/2</p> Signup and view all the answers

What is the value of 'Cos 30'?

<p>√3/2</p> Signup and view all the answers

What is the value of 'Cos 60'?

<p>0.5</p> Signup and view all the answers

What is the value of 'Cos 45'?

<p>√2/2</p> Signup and view all the answers

What is the value of 'Tan 30'?

<p>√3/3</p> Signup and view all the answers

What is the value of 'Tan 60'?

<p>√3</p> Signup and view all the answers

What is the value of 'Tan 45'?

<p>1</p> Signup and view all the answers

What does 'Standard Position' refer to?

<p>When its vertex is at the origin and one ray is on the positive x-axis.</p> Signup and view all the answers

What is the 'Initial Side' of an angle?

<p>Angle on the ray on the x-axis</p> Signup and view all the answers

What is the 'Terminal Side' of an angle?

<p>The ray other than the initial side.</p> Signup and view all the answers

What are 'Coterminal Angles'?

<p>Angles in standard position with the same terminal side.</p> Signup and view all the answers

What is a 'Reference Angle'?

<p>Positive acute angle formed by the terminal side of theta and the x-axis</p> Signup and view all the answers

What is a 'Radian'?

<p>Unit of angle measure based on arc length.</p> Signup and view all the answers

What is the conversion from degrees to radians?

<p>Multiply by π/180</p> Signup and view all the answers

What is the conversion from radians to degrees?

<p>Multiply by 180/Ï€</p> Signup and view all the answers

What is a 'Unit Circle'?

<p>Circle with a radius of 1 unit.</p> Signup and view all the answers

What is the 'Arc Length Formula'?

<p>s = rθ</p> Signup and view all the answers

What are 'Inverse Trig Functions'?

<p>Functions that reverse the effect of the original trigonometric functions.</p> Signup and view all the answers

What is the 'Area of A Triangle' formula?

<p>Area = 1/2 * base * height</p> Signup and view all the answers

What is the 'Law of Sines'?

<p>Applies to any triangle, relates the ratios of the sides to the sine of their opposite angles.</p> Signup and view all the answers

What is the 'Law of Cosines'?

<p>Relates the lengths of sides of a triangle with the cosine of one of its angles.</p> Signup and view all the answers

What is 'Heron's Formula'?

<p>Used to find the area of a triangle when the lengths of all three sides are known.</p> Signup and view all the answers

What are 'Transformation of Sin & Cos Graphs'?

<p>Shifting, reflection, and stretching of sine and cosine functions.</p> Signup and view all the answers

What are 'Transformation of Tangent Graphs'?

<p>Shifting, reflection, and stretching of tangent functions.</p> Signup and view all the answers

What are 'Transformation of Cotangent Graphs'?

<p>Shifting, reflection, and stretching of cotangent functions.</p> Signup and view all the answers

What are 'Reciprocal Identities'?

<p>Relationships that show the reciprocals of sine, cosine, and tangent.</p> Signup and view all the answers

What are 'Tangent and Cotangent Ratio Identities'?

<p>Relationships between tangent and cotangent in terms of sine and cosine.</p> Signup and view all the answers

What are 'Pythagorean Identities'?

<p>Relations involving squares of sine and cosine.</p> Signup and view all the answers

What are 'Negative-Angle Identities'?

<p>Identities that express trigonometric functions for negative angles.</p> Signup and view all the answers

What are 'Sum and Difference Identities'?

<p>Formulas for finding sine, cosine, or tangent of the sum or difference of two angles.</p> Signup and view all the answers

What are 'Double Angle Identities'?

<p>Formulas for finding trigonometric functions of double angles.</p> Signup and view all the answers

What are 'Half Angle Identities'?

<p>Formulas for finding trigonometric functions of half angles.</p> Signup and view all the answers

Flashcards

Sine

Opposite side length divided by hypotenuse length in a right triangle.

Cosine

Adjacent side length divided by hypotenuse length in a right triangle.

Tangent

Ratio of the opposite side to the adjacent side in a right triangle; sine divided by cosine.

Sine 30°

Equals 1/2

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Sine 60°

Equals √3/2.

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Sine 45°

Equals √2/2

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Cosine 30°

Equals √3/2.

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Cosine 60°

Equals 1/2.

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Cosine 45°

Equals √2/2.

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Tangent 30°

Equals √3/3.

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Tangent 60°

Equals √3

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Tangent 45°

Equals 1

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Standard Position (Angle)

An angle with its vertex at the origin and one ray along the positive x-axis.

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Initial Side

The side of an angle that lies along the x-axis in standard position.

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Terminal Side

The ray resulting from the rotation of the initial side.

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Coterminal Angles

Angles sharing the same terminal side. Add/subtract 360° or 2π.

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Reference Angle

The smallest positive angle between the terminal side and the x-axis.

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Radian

Angular measure based on the radius of a circle.

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Degrees to Radians

Multiply degrees by π/180.

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Radians to Degrees

Multiply radians by 180/Ï€.

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Unit Circle

A circle with radius 1, used to define trigonometric functions.

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Arc Length Formula

Relates angle, radius, and arc length: s = rθ.

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Area of a Triangle

Various formulas based on known triangle variables.

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Law of Sines

Relates sides to sines of opposite angles.

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Law of Cosines

Relates the sides of a triangle to the cosine of one of its angles.

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Heron’s Formula

Calculates triangle area when all three sides are known.

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Transformation of Sin & Cos Graphs

Stretching, compressing, or shifting sine and cosine functions.

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Transformation of Tangent Graphs

Changes affect period and asymptotes.

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Reciprocal Identities

Defines relationships between trig functions and their reciprocals.

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Tangent and Cotangent Ratio Identities

tan(x) = sin(x)/cos(x), cot(x) = cos(x)/sin(x)

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Study Notes

Sine, Cosine, and Tangent

  • Sine: A trigonometric function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
  • Cosine: Defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
  • Tangent: The ratio of the sine to the cosine, or equivalently, the ratio of the opposite side to the adjacent side in a right triangle.

Key Sine Values

  • Sine 30: Equals 1/2.
  • Sine 60: Equals √3/2.
  • Sine 45: Equals √2/2.

Key Cosine Values

  • Cosine 30: Equals √3/2.
  • Cosine 60: Equals 1/2.
  • Cosine 45: Equals √2/2.

Key Tangent Values

  • Tangent 30: Equals √3/3.
  • Tangent 60: Equals √3.
  • Tangent 45: Equals 1.

Angle Definitions

  • Standard Position: An angle positioned with its vertex at the origin and one ray along the positive x-axis.
  • Initial Side: The side of an angle that lies along the x-axis in standard position.
  • Terminal Side: The ray that results from the rotation of the initial side.

Special Angle Concepts

  • Coterminal Angles: Angles that share the same terminal side when in standard position; can be obtained by adding or subtracting multiples of 360° or 2Ï€ radians.
  • Reference Angle: The smallest positive angle formed between the terminal side of an angle and the x-axis.

Unit of Measure

  • Radian: A unit of angular measure based on the radius of a circle used primarily in trigonometry.
  • Degrees to Radians: Conversion requires multiplying degrees by Ï€/180.
  • Radians to Degrees: Conversion requires multiplying radians by 180/Ï€.

Key Trigonometric Concepts

  • Unit Circle: A circle with a radius of one, used to define trigonometric functions.
  • Arc Length Formula: Relates the angle in radians and the radius to find the length of the arc.

Areas and Laws

  • Area of a Triangle: Can be calculated using various formulas depending on known variables.
  • Law of Sines: Relates the ratios of the lengths of sides of a triangle to the sines of the corresponding opposite angles.
  • Law of Cosines: Relates the lengths of sides of a triangle to the cosine of one of its angles.
  • Heron’s Formula: Calculates the area of a triangle when the lengths of all three sides are known.

Graph Transformations

  • Transformation of Sin & Cos Graphs: Involves stretching, compressing, or shifting the sine and cosine functions.
  • Transformation of Tangent Graphs: Similar transformations apply but will modify the periodic behavior.
  • Transformation of Cotangent Graphs: Affects the position and shape of the cotangent function graph.

Trigonometric Identities

  • Reciprocal Identities: Define relationships between the primary trigonometric functions and their reciprocals.
  • Tangent and Cotangent Ratio Identities: Establish relationships between tangent, cotangent, sine, and cosine.
  • Pythagorean Identities: Derived from the Pythagorean theorem and relate the squares of sine, cosine, and tangent.
  • Negative-Angle Identities: Allow calculations of functions of negative angles.
  • Sum and Difference Identities: Enable the calculation of trigonometric functions for the sum or difference of angles.
  • Double Angle Identities: Provide formulas to express trigonometric functions of double angles.
  • Half Angle Identities: Provide formulas for trigonometric functions of half angles.

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Test your knowledge of trigonometric functions with these flashcards. Each card covers key terms and values related to sine and cosine, including specific angles. Perfect for students looking to reinforce their understanding of trigonometry in Algebra 3.

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