Algebra 2B - Unit 3: Trigonometry Flashcards
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Questions and Answers

What are special right triangles?

special right triangles

What ratios can be used to find the missing side lengths of a 45-45-90 triangle?

Refer to the provided image.

What are the lengths of sides a and b in the triangle?

a = 5 ft, b = 5 ft

What are the lengths of sides b and c in the triangle?

<p>b = 3 in, c = 3√2 in</p> Signup and view all the answers

What are the lengths of sides a and b in the triangle?

<p>a = √2/2 m, b = √2/2 m</p> Signup and view all the answers

What is the sine of angle a?

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

What is the cosine of angle a?

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

What is the tangent of angle a?

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

What ratios can be used to find the missing side lengths of a 30-60-90 triangle?

<p>Refer to the provided image.</p> Signup and view all the answers

What are the lengths of sides a and b in the triangle?

<p>a = 8√3 in, b = 8 in</p> Signup and view all the answers

What are the lengths of sides a and b in the triangle?

<p>a = 1/2 cm, b = √3/2 cm</p> Signup and view all the answers

What are the lengths of sides b and c in the triangle?

<p>b = 7 ft, c = 14 ft</p> Signup and view all the answers

What are the sine values of the 30-degree angle and the 60-degree angle?

<p>Sine of 30 degrees is 1/2, sine of 60 degrees is √3/2</p> Signup and view all the answers

What are the cosine values of the 30-degree angle and the 60-degree angle?

<p>Cosine of 30 degrees is √3/2, cosine of 60 degrees is 1/2</p> Signup and view all the answers

What are the tangent values of the 30-degree angle and the 60-degree angle?

<p>Tangent of 30 degrees is √3/3, tangent of 60 degrees is √3</p> Signup and view all the answers

What is the sine of the 30 angle and the sine of the 60 angle?

<p>Sine of 30 = 1/2, Sine of 60 = √3/2</p> Signup and view all the answers

What is the cosine of the 30 angle and the cosine of the 60 angle?

<p>Cosine of 30 = √3/2, Cosine of 60 = 1/2</p> Signup and view all the answers

What is the measure of a 75° angle in radians?

<p>5π/12</p> Signup and view all the answers

What is the measure of a 135° angle in radians?

<p>3π/4</p> Signup and view all the answers

If an angle measures 5π/4 radians, what is the measure in degrees?

<p>225°</p> Signup and view all the answers

Which angles are coterminal with a 140° angle? Select all that apply.

<p>580°</p> Signup and view all the answers

Which angles are coterminal with a 30° angle? Select all that apply.

<p>-330°</p> Signup and view all the answers

What is the tangent of the angle?

<p>tan(o) = -1</p> Signup and view all the answers

What is the sine of the angle?

<p>sin(o) = -√3/2</p> Signup and view all the answers

What is the cosine of the angle?

<p>cos(o) = -√3/2</p> Signup and view all the answers

What is the measure of an angle in standard position that has a terminal side from the origin to point E?

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

Which point forms a terminal side with the origin that creates an angle with a sine of 0 and a cosine of -1?

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

Which point forms a terminal side with the origin that creates an angle with a sine of 1/2 and a cosine of √3/2?

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

What is the tangent of the angle?

<p>tan(o) = -8/15</p> Signup and view all the answers

What is the measure of an angle in standard position that has a terminal side from the origin to point i?

<p>5π/6</p> Signup and view all the answers

Which statements are true regarding the trigonometric ratios of this triangle? Select all that apply.

Signup and view all the answers

What are the special right triangles mentioned in lesson 12?

<p>Special right triangles are the 45-45-90 and 30-60-90 triangles.</p> Signup and view all the answers

What are the correct ratios for a 45-45-90 triangle?

<p>1:1:√2</p> Signup and view all the answers

If the lengths of sides a and b are both 5 ft, what type of triangle is it?

<p>It is an isosceles triangle.</p> Signup and view all the answers

What are the lengths of sides b and c if b = 3 in?

<p>b = 3 in, c = 3√2 in.</p> Signup and view all the answers

What are the lengths of sides a and b if a = √2/2 m?

<p>a = √2/2 m, b = √2/2 m.</p> Signup and view all the answers

What is the sine of angle a at 45 degrees?

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

What is the cosine of angle a at 45 degrees?

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

What is the tangent of angle a at 45 degrees?

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

What are the correct ratios for a 30-60-90 triangle?

<p>1:√3:2</p> Signup and view all the answers

If a = 8√3 in, what is the length of side b in a 30-60-90 triangle?

<p>b = 8 in.</p> Signup and view all the answers

If a = 1/2 cm in a 30-60-90 triangle, what is the length of side b?

<p>b = √3/2 cm.</p> Signup and view all the answers

If b = 7 ft, what is the length of side c in a right triangle?

<p>c = 14 ft.</p> Signup and view all the answers

What is the sine of the 30-degree angle?

<p>1/2</p> Signup and view all the answers

What is the sine of the 60-degree angle?

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

What is the cosine of the 30-degree angle?

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

What is the cosine of the 60-degree angle?

<p>1/2</p> Signup and view all the answers

What is the tangent of the 30-degree angle?

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

What is the tangent of the 60-degree angle?

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

What angle corresponds to 𝝅 radians?

<p>Angle ace.</p> Signup and view all the answers

What is the measure of angle acd in radians?

<p>𝝅/2.</p> Signup and view all the answers

If o equals 45°, what is the measure of arc ab in radians?

<p>𝝅/4.</p> Signup and view all the answers

What is the measure of a 75° angle in radians?

<p>5𝝅/12.</p> Signup and view all the answers

What is the measure of a 135° angle in radians?

<p>3𝝅/4.</p> Signup and view all the answers

If an angle measures 5𝝅/4 radians, what is the measure in degrees?

<p>225°.</p> Signup and view all the answers

If an angle measures 2𝝅/9 radians, what is the measure in degrees?

<p>40°.</p> Signup and view all the answers

Which angles are coterminal with a 140° angle? (Select all that apply)

<p>580°</p> Signup and view all the answers

Which angles are coterminal with a 95° angle? (Select all that apply)

<p>-265°</p> Signup and view all the answers

Which angle is coterminal with an angle that measures 𝝅/4 radians?

<p>9𝝅/4.</p> Signup and view all the answers

Which angle is coterminal with an angle that measures 2𝝅/3 radians?

<p>-4𝝅/3.</p> Signup and view all the answers

If an angle measures 6𝝅/5 radians, what is the measure in degrees?

<p>216°.</p> Signup and view all the answers

Which angles are coterminal with a 30° angle? (Select all that apply)

<p>-330°</p> Signup and view all the answers

What is the radian measure of positive angle acb?

<p>𝝅/4.</p> Signup and view all the answers

What point forms a terminal side that creates an angle with a sine of 0 and a cosine of -1?

<p>j.</p> Signup and view all the answers

What is the tangent of the angle represented by (1/2, -√3/2)?

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

What is the sine of the angle represented by (-√3/2, 1/2)?

<p>1/2.</p> Signup and view all the answers

Use reference angles to find the sine of 210°.

<p>-1/2.</p> Signup and view all the answers

Use reference angles to find the cosine of 210°.

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

Use reference angles to find the tangent of 225°.

<ol> <li></li> </ol> Signup and view all the answers

What mistake did Yelenia make, if any?

<p>In step 4, she used the incorrect definition of sine.</p> Signup and view all the answers

What is the best critique of Thomas's proof?

<p>Thomas should not have chosen π/4 as the angle measurement.</p> Signup and view all the answers

If angle o has a terminal side in quadrant ii and sin(o) = 24/25, what is cos(o)?

<p>-7/25.</p> Signup and view all the answers

From the previous question, what is tan(o)?

<p>-24/7.</p> Signup and view all the answers

If angle o has a terminal side in quadrant iii and cos(o) = -4/5, what is tan(o)?

<p>3/4.</p> Signup and view all the answers

Study Notes

Special Right Triangles

  • Key formulas for 45-45-90 triangles: sides are in the ratio 1:1:√2.
  • For a 45-45-90 triangle with each leg of length 5 ft, hypotenuse is 5√2 ft.
  • In a 30-60-90 triangle, the sides are in the ratio 1:√3:2.
  • Example dimensions: for 30-60-90 triangle with shorter leg of 3 in, the longer leg is 3√3 in, and the hypotenuse is 6 in.

Trigonometric Functions

  • Sine, cosine, and tangent for angle 45°: sin(45°) = cos(45°) = √2/2 and tan(45°) = 1.
  • For angle 30°: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = √3/3.
  • For angle 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.

Radian Measure

  • A full circle is 2π radians; hence angle acf at 3/4 of a circle measures 3π/4.
  • π radians corresponds to a 180° angle; angles can be converted between degrees and radians using multiplication by (π/180) or (180/π).
  • Examples: 75° = 5π/12 radians and 135° = 3π/4 radians.

Coterminal Angles

  • Angles that share the same terminal side can be found by adding or subtracting multiples of 360° or 2π radians.
  • Examples of coterminal angles for 140°: 500°, 580°; for 95°: 815°, -265°.

Quadrants and Reference Angles

  • Reference angles help find the sine, cosine, and tangent for angles in different quadrants.
  • In quadrant II: sin(120°) = √3/2, cos(120°) = -1/2, tan(120°) = -√3.
  • In quadrant III: sin(210°) = -1/2, cos(210°) = -√3/2, tan(210°) = √3/3.

Pythagorean Identity

  • The identity states that sin²(o) + cos²(o) = 1 holds true for any angle.
  • Example: if sin(o) = 24/25, then cos(o) = -7/25 based on sinusoidal relationships defined by the unit circle.

Unit Circle Properties

  • The unit circle relates the sine and cosine of angles to coordinates of points (x, y) where x = cos(θ) and y = sin(θ).
  • The distance of a point from the origin (1 in the unit circle) relates to hypotenuse definition in sine and cosine: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse.

Angle Conversions

  • Angle conversions between degrees and radians involve multiplying the degree measure by π/180.
  • Example conversions: 6π/5 radians = 216° and 5π/8 radians = 112.5°.

Key Values of Trigonometric Functions

  • Common angles have defining sine and cosine values:
    • sin(45°) = √2/2
    • sin(225°) = -√2/2, cos(225°) = -√2/2, tan(225°) = 1.

General Conclusions

  • Understanding special right triangles, angles in both degree and radian measures, and properties of the unit circle are fundamental in trigonometry.
  • Mastery of reference angles and trigonometric identities aids in solving a variety of problems related to angles and triangles.

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Description

This quiz covers special right triangles, including the properties and ratios of 45-45-90 triangles. Examine key definitions and apply them to determine side lengths in various scenarios. Test your understanding of trigonometric principles presented in this unit.

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