Plane trigonometry focuses on angles that lie on a two-dimensional field. Which description matches this focus?

Circular functions are described in this section as ratios between the lengths of the sides of a triangle, commonly approached using a right triangle within a unit circle. Which term refers to these functions?

In the rectangular coordinate system, the horizontal axis is called the ______ and the vertical axis is called the ______.

Identify the quadrant for a point with coordinates (x > 0, y < 0). Use the quadrant numbering I to IV counterclockwise starting from the positive x- and y-axes.

The coordinates of a point P are written as an ordered pair (x, y). Which term is the directed distance of the point from the y-axis, and which term is the directed distance from the x-axis?

Consider the list of points: A(3, 2), B(0, −3), C(−4, 5), D(2, 0), E(3, −4), F(−2, −3). Which point lies on the y-axis?

DoK 1. Recall: In the proof diagram, what is the length of the horizontal segment between P1 and P3?

DoK 2. Application: Using the exercise grid with labeled points A, B, C, D, E, F, G, H, I and axes marked x and y, identify the coordinates of point G, located where the vertical through y intersects the horizontal through x.

DoK 3. Strategic Thinking: Suppose point C lies on the x-axis to the left of the origin and point F lies on the x-axis to the right of the origin at an equal number of grid units from the origin. If the origin is at G, which statement about the distances GC and GF is correct?

Compute the distance between points A(3, 4) and B(−1, 3). Use the distance formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ .

Compute the distance between points C(0, −3) and D(4, 0). Use the distance formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ .

Compute the distance between points E(−2, 2) and F(5, 2). Use the distance formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ .

Compute the distance between points I(7, −3) and J(−2, −1). Use the distance formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ .

Compute the distance between points K(3, −7) and L(−5, −1). Use the distance formula $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ .

According to the angular measure definition, how many degrees are in one revolution?

Using the relationship between degrees and radians for a unit circle, which equality is correct?

Which statement best expresses the proportionality between central angles and their intercepted arcs in a circle?

A circle of radius r has central angles θ1 = 1 rad and θ2. If s1 and s2 are the corresponding arc lengths, which relationship holds?

According to the theorem on arc length, what is the formula for the length S of an arc intercepted by a central angle θ radians in a circle of radius r?

Two arcs s1 and s2 in the same circle correspond to central angles θ1 and θ2. If s1:s2 = 2:3, what is θ1:θ2?

In the diagram of a circle with two central angles θ1 and θ2 intercepting arcs s1 and s2, respectively, which equality is illustrated?

Recall the formula for the length of an arc intercepted by a central angle θ (in radians) in a circle of radius r.

Using S = r θ, compute the arc length when r = 160 cm and θ = 0.12 rad. Give your answer to the nearest tenth of a centimeter.

A circle has radius 9 cm and central angle 72°. Which expression correctly converts the angle to radians before using S = r θ?

Compute the arc length for r = 9 cm and θ = 72° using S = r θ with θ in radians. Round to two decimal places.

How many degrees are between the hands of a clock at 5:00? Assume the hour hand points exactly at 5 and the minute hand at 12.

Through how many degrees does the minute hand of a clock move in 40 minutes? Use 360° per 60 minutes.

A clock’s minute hand is 10 cm long. Find the distance the tip travels from 12:00 to 12:45. Use arc length S = rθ, with θ = 3/4 of a full circle.

The diameter of a jeep wheel is 0.8 m. How many radians does a spoke rotate when the jeep travels 1 km, assuming no slip? Use S = rθ with r = 0.4 m.

An equilateral triangle has side 15 cm and is centered on a circle so that two vertices lie on the circle and the third is at the center. Which statement correctly describes the difference between the length of the chord and its intercepted arc?

Refer to the diagram showing rays OA along the positive x-axis, and rays OB and OC in Quadrants I and II respectively. Which angle belongs to Quadrant II?

In trigonometry, what is meant by the terminal side of an angle?

An angle in standard position measures 675°. Which of the following is a positive co-terminal angle less than 360°?

Using the unit circle definition, which expression equals sin t for a point P(x, y) on the unit circle corresponding to angle t?

For P(x, y) on the unit circle, which trigonometric function is defined as x/y?

Given P(x, y) lies on the unit circle, select the correct definition of sec t.

If P(x, y) is on the unit circle with y = −1/2, what is csc t?

The distance of point P from the origin on the unit circle is called the radius vector r. Which statement is correct?

Given an angle θ with terminal side through P(3, −4), which expression correctly gives tan θ using terminal side coordinates?

Apply concepts: For θ in standard position with terminal side through B(12, −9), compute sec θ.

Strategic thinking: For θ in standard position with terminal side through C(−24, −7), decide the signs of sin θ, cos θ, and tan θ and select the correct set.

Recall: In Quadrant II, which trigonometric functions are positive? Choose the best answer.

Recall: In Quadrant III, which trigonometric functions are positive?

Recall: In Quadrant IV, which single basic trigonometric function is positive?

Skill: Determine the sign pattern across quadrants. Which pair is negative in Quadrant I?

Strategic: Given cot θ = −3/4 and sin θ is positive, which quadrant contains θ?

Strategic: Given cot θ = −3 and θ is in Quadrant IV, find tan θ.

Strategic: Given tan θ = −24/7 and cos θ is negative, identify the quadrant of θ.

Skill: Given sec θ = −15/9 and θ is in Quadrant III, what is cos θ?

Strategic: Given sin θ = −4/√65 and tan θ is negative, determine the sign of cos θ.

Skill: Given tan θ = 3 and sin θ > 0, which quadrant contains θ?

Strategic: Given cot θ = −2 and sec θ > 0, find the quadrant of θ.

Find the coordinates of the point on the unit circle corresponding to θ = 3π/7. Express your answer in simplest fractional or radical form as an ordered pair (x, y).

Given sin θ = −3/4 and sec θ = 4√7/7, determine cos θ.

Given cot θ = −4/3 and sin θ = 3/5, determine cos θ.

Given cos θ = −1/2 and csc θ = 2√3/3, determine tan θ.

Given sin θ = −1/3 and tan θ = √2/4, determine cos θ.

Given csc θ = −√7/2 and tan θ = −2/√3, determine sin θ.

Evaluate: sin θ + cos θ cot θ.

Simplify: sec θ − sin θ tan θ.

Simplify: 2 cos² θ − sin² θ + 1 all over cos θ.

Evaluate: sin² θ/(1 − 2 sin² θ).

Simplify: cos θ (sec θ + tan θ) − sin θ.

Simplify: sin θ (sin θ + cot θ) − cos θ.

Evaluate the statement: 1 − sin² 270° = cos² 270°. Select the correct judgment based on Pythagorean identities.

Which formula from the material correctly expresses functions of negative angles for tangent?

Given the diagram showing angle A in quadrant III, which expression matches its reference angle according to the rule provided?

Decide whether the equality is true: sin 90° = sin 45° cos 45°.

Select the correct identity for cotangent of a negative angle from the provided table.

Strategic reasoning: Use the identities for negative angles to simplify the expression sin(−θ) + cos(−θ) when sin θ = 3/5 and cos θ = 4/5. Choose the correct value.

Prove or disprove by evaluation: 2 sin 45° = cos 45° + sin 45°. Choose the correct conclusion.

Recall: For angles in quadrant IV, what is the reference angle θ in terms of A?

Using the unit-circle diagram showing angle −θ in standard position, which identity is correct for the sine of a negative angle?

From the identities shown beside the unit-circle diagram, which is the correct expression for cos(−θ)?

According to the diagrammed identities, what is tan(−θ) in terms of tan θ?

Which negative-angle identity is correct for cotangent, based on the visual and formulas provided?

Which identity holds for csc(−θ)?

Evaluate sec 225° − sin 225° • tan 225°. Use exact values for trigonometric functions at 225°.

Compute sin 210° + tan 210°. Provide the exact value.

Simplify cot 135° − tan 135°. Choose the exact simplified value.

Find the exact value of 1 − 2 cos² 240°.

Compute sin 135°(1 − cot 135°). Give the exact value.

Evaluate tan 135° − sin 135°.

Determine the value of cos 330° − sin 330°. Use exact values.

Evaluate sin 120° cos 150° − sin 240° sin 330°. Use exact values of special angles.

Solve for the condition on θ: sin θ = cos θ √sec θ − 1. For which θ in [0°, 360°) does this hold?

Use quotient and reciprocal identities to simplify the expression: (tan 330° + sec 330° • sin 330°) + 1.

Evaluate sec² 330°(1 + cos 330° • sin 330°) − (tan 330° + sec 330°)² + 1 using identities and exact angle values.

Recall the fundamental Pythagorean identity connecting sine and cosine. Which equation is correct?

Select the identity that expresses $$1 + tan^2 \theta$$ in terms of $$sec \theta$$ .

If csc x = 5 and cos x > 0, determine the quadrant of x and the sign of sin x.