Unit 6 Trigonometry

Open Ended

Why do you think trigonometry and the unit circle are important in advanced algebra and real-world applications?

Multiple Choice

Which of the following is NOT a basic trigonometric ratio?

• Sine

• Cosine

• Tangent

• Secant

Multiple Choice

Which of the following is a learning target for this lesson?

• I can define sine, cosine, and tangent ratios using all four quadrants of the unit circle.

• I can solve quadratic equations by factoring.

• I can graph linear equations.

• I can find the area of a triangle.

Open Ended

Based on the diagrams and definitions, what is the difference between the initial side and the terminal side of an angle in standard position?

Multiple Choice

Which of the following statements about positive, negative, and coterminal angles is correct?

• Positive angles are measured counterclockwise, negative angles are measured clockwise, and coterminal angles share the same initial and terminal sides.

• Positive angles are measured clockwise, negative angles are measured counterclockwise, and coterminal angles have different terminal sides.

• Coterminal angles never share the same initial side.

• Negative angles are always larger than positive angles.

Multiple Select

Select all the angles that are coterminal with a 45° angle.

• 405°

• -315°

• 135°

• 660°

Open Ended

What is a radian, and how is it defined in terms of a circle's radius and arc length?

Multiple Choice

What is the definition of a radian, and how is it related to the radius and arc length of a circle?

• A radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

• A radian is the distance from the center of a circle to its circumference.

• A radian is the area of a sector with a radius of 1.

• A radian is the circumference of a circle divided by its diameter.

Multiple Choice

Explain the relationship between radians and degrees, and describe how many radians are in a full revolution of a circle.

• A full revolution is 2π radians, which is equal to 360 degrees.

• A full revolution is π radians, which is equal to 180 degrees.

• A full revolution is 4π radians, which is equal to 720 degrees.

• A full revolution is π/2 radians, which is equal to 90 degrees.

Multiple Choice

What is the equivalent degree measure for the angle \( \frac{2\pi}{3} \) radians?

• 120°

• 90°

• 180°

• 60°

Multiple Select

Which of the following are true about the unit circle?

• It has a radius of 1.

• It is centered at the point (0,0).

• It has a circumference of 2π.

• It has a diameter of π.

Open Ended

Describe the key features of the unit circle and explain why it is important in trigonometry.

Multiple Choice

Which of the following statements best describes how sine, cosine, and tangent can be defined using the unit circle and right triangle trigonometry?

• They are defined only for angles in the first quadrant.

• They are defined using the lengths of the sides of a right triangle inscribed in the unit circle.

• They are defined using the area of the triangle formed by the radius and the x-axis.

• They are defined using the circumference of the unit circle.

Multiple Choice

What are the definitions of sine, cosine, and tangent in terms of the sides of a right triangle?

• Sine = adjacent/hypotenuse, Cosine = opposite/hypotenuse, Tangent = adjacent/opposite

• Sine = opposite/hypotenuse, Cosine = adjacent/hypotenuse, Tangent = opposite/adjacent

• Sine = hypotenuse/opposite, Cosine = hypotenuse/adjacent, Tangent = adjacent/opposite

• Sine = opposite/adjacent, Cosine = adjacent/opposite, Tangent = hypotenuse/opposite

Multiple Select

Select all correct statements about the relationship between the coordinates (x, y) on the unit circle and the trigonometric functions sine, cosine, and tangent.

• cosθ = x

• sinθ = y

• tanθ = x/y

• tanθ = y/x

Open Ended

How do the definitions of sine, cosine, and tangent for a right triangle relate to their definitions on the unit circle?

Multiple Choice

Which of the following statements about the signs of x and y coordinates in the four quadrants is correct?

• In quadrant II, both x and y are positive.

• In quadrant III, x is positive and y is negative.

• In quadrant IV, x is positive and y is negative.

• In quadrant I, x is negative and y is positive.

Multiple Choice

Will tan(330°) be positive or negative?

• Positive

• Negative

Multiple Choice

Will sin(5π/6) be positive or negative?

• Positive

• Negative

Multiple Choice

Will the value of tan(4π/3) be positive or negative?

• Positive

• Negative

Multiple Choice

Will the value of cos(120°) be positive or negative?

• Positive

• Negative

Open Ended

Explain how the reference angle is determined for an angle in standard position. Why is the reference angle important in trigonometry?

Multiple Choice

What is the relationship between the reference angle α and θ in degrees for an angle in Quadrant II?

• α = θ

• α = 180° − θ

• α = θ − 180°

• α = 360° − θ

Open Ended

Trigonometric ratios are used to relate the angles of a triangle to the lengths of its sides. Can you think of a real-world situation where understanding trigonometric ratios would be useful?

Open Ended

What questions do you still have about trigonometric ratios or the unit circle after today's lesson?