D1.3 Find two angles with the same trigonometric ratio

Ontario Grade 11 Mathematics

Trigonometric Functions

Finding All Angles That Share the Same Sine, Cosine, or Tangent Value (MCR3U – D1.3)

You already know how to find the sine, cosine, or tangent of a single angle. But here is a twist: if someone tells you that sin θ = 0.6, can you find every angle between 0° and 360° that makes that statement true? It turns out there are always two such angles (for most ratio values), and knowing how to locate both of them is an essential skill in Grade 11 trigonometry. This lesson builds directly on your Grade 10 work with the unit circle, reference angles, and the signs of trigonometric ratios in each quadrant. By the end, you will have a reliable, step-by-step method for tracking down both angles every time.

What you will learn

Explain why two different angles between 0° and 360° can share the same trigonometric ratio.
Use the CAST rule to identify which quadrants produce a positive or negative ratio.
Find the reference angle from a given trigonometric ratio.
Determine both angles in the interval 0° to 360° that share a given sine, cosine, or tangent value.
Verify both answers by checking their signs and reference angles.

Prerequisite Bridge: The Unit Circle and Quadrants

Picture a circle with a radius of 1 unit centred at the origin of an x-y grid. An angle θ is measured counterclockwise from the positive x-axis. The point where the terminal arm of θ meets the circle has coordinates (cos θ, sin θ). This is the unit-circle idea from Grade 10, and it is the foundation for everything in this lesson.

The x-y plane is divided into four quadrants. Quadrant I is top-right (both x and y positive). Quadrant II is top-left (x negative, y positive). Quadrant III is bottom-left (both negative). Quadrant IV is bottom-right (x positive, y negative). Because cos θ equals the x-coordinate and sin θ equals the y-coordinate of the point on the unit circle, the signs of these ratios depend entirely on which quadrant the terminal arm lands in.

The tangent ratio is defined as tan θ = sin θ ÷ cos θ, so its sign follows from the signs of sin and cos. When sin and cos share the same sign, tan is positive; when they differ in sign, tan is negative.

\tan\theta = \frac{\sin\theta}{\cos\theta}

Angles are measured counterclockwise from the positive x-axis, from 0° to 360°.
The coordinates of the unit-circle point are (cos θ, sin θ).
The sign of each ratio depends on the quadrant of the terminal arm.
tan θ = sin θ ÷ cos θ, so its sign is determined by the signs of sin and cos.

The CAST Rule: Which Ratios Are Positive Where?

A quick memory tool called the CAST rule tells you which primary trigonometric ratio is positive in each quadrant. Reading counterclockwise from Quadrant IV: C (Cosine positive), A (All positive), S (Sine positive), T (Tangent positive). You can also read it clockwise from Quadrant I as All, Sine, Tangent, Cosine — whatever helps you remember.

In Quadrant I, all three ratios are positive because both coordinates are positive. In Quadrant II, only sine is positive (y is positive, x is negative). In Quadrant III, only tangent is positive (both coordinates are negative, so sin÷cos gives a positive result). In Quadrant IV, only cosine is positive (x is positive, y is negative).

Knowing the sign of a given ratio immediately tells you which two quadrants to look in. For example, a positive sine value means the terminal arm is in Quadrant I or Quadrant II. A negative cosine value means the terminal arm is in Quadrant II or Quadrant III. This narrows your search before you calculate anything.

CAST: Cosine (+) in Q IV, All (+) in Q I, Sine (+) in Q II, Tangent (+) in Q III.
A positive ratio → terminal arm in the two quadrants where that ratio is positive.
A negative ratio → terminal arm in the two quadrants where that ratio is negative.
Identifying the correct pair of quadrants is the first step in every problem.

Reference Angles: The Bridge Between Quadrants

A reference angle is the acute angle (between 0° and 90°) formed between the terminal arm of θ and the nearest part of the x-axis. It is always positive. The key fact is this: all four angles that share the same reference angle have trigonometric ratios with the same absolute value — only the signs differ based on the quadrant.

To find the reference angle α from a given ratio, ignore the negative sign (if any) and apply the inverse trigonometric function on your calculator. For example, if sin θ = −0.74, you find α = sin⁻¹(0.74) ≈ 47.7°. That 47.7° is the reference angle. You then place it in the correct quadrants using the CAST rule.

Once you have the reference angle α, you convert it to the actual angle θ in each relevant quadrant using a simple formula. The conversion formulas are: Quadrant I: θ = α. Quadrant II: θ = 180° − α. Quadrant III: θ = 180° + α. Quadrant IV: θ = 360° − α. These formulas come directly from the symmetry of the unit circle — they are not arbitrary rules to memorise blindly, but a reflection of how angles relate to the x-axis in each part of the plane.

\theta_{\text{QII}} = 180^\circ - \alpha, \quad \theta_{\text{QIII}} = 180^\circ + \alpha, \quad \theta_{\text{QIV}} = 360^\circ - \alpha

The reference angle α is the acute angle between the terminal arm and the nearest x-axis.
Find α by applying the inverse trig function to the absolute value of the ratio.
Q I: θ = α. Q II: θ = 180° − α. Q III: θ = 180° + α. Q IV: θ = 360° − α.
Both target angles share the same reference angle and have the same absolute ratio value.

Putting It All Together: A Four-Step Method

Here is a reliable four-step method you can apply to any problem of this type. Step 1 — Identify the ratio and its sign. Note which trigonometric function is given and whether its value is positive or negative. Step 2 — Use the CAST rule to find the two quadrants where that ratio has the correct sign. Step 3 — Find the reference angle α by entering the absolute value of the ratio into the appropriate inverse function on your calculator. Step 4 — Use the quadrant-conversion formulas to write both angles, and then verify each by checking its sine, cosine, or tangent on a calculator.

A small but important note: the angles 0°, 90°, 180°, 270°, and 360° are boundary cases. At these angles, one of the coordinates on the unit circle is zero, which causes tan to be either zero or undefined. These boundary cases can be handled directly without a reference angle — just recall the exact values from the unit circle.

Always verify your two answers at the end. Substitute each angle back into the original ratio on your calculator. Both results should match the given value (to the decimal places shown). This habit catches sign errors and quadrant mistakes before they cost you marks.

Step 1: Note the trig function and the sign of the given ratio.
Step 2: CAST rule → identify the two correct quadrants.
Step 3: Reference angle α = inverse trig function of the absolute ratio value.
Step 4: Convert α to θ in each quadrant, then verify both answers on a calculator.

Special Exact Values and Boundary Cases

Some trigonometric ratios produce exact reference angles that you should recognise without a calculator: sin 30° = cos 60° = 0.5, sin 45° = cos 45° = 1/√2 ≈ 0.7071, sin 60° = cos 30° = √3/2 ≈ 0.8660, and tan 45° = 1. When a problem gives you one of these values, use the exact angle rather than a rounded decimal.

For example, if cos θ = −0.5, the reference angle is exactly 60° (since cos 60° = 0.5). Cosine is negative in Quadrants II and III, so the two angles are 180° − 60° = 120° and 180° + 60° = 240°. You can verify: cos 120° = −0.5 and cos 240° = −0.5. Both correct.

Knowing these special values also helps you spot errors. If your calculator gives a reference angle of 29.9° when you expected 30°, that is a rounding issue, not a different angle — round sensibly and state the exact value when it is clean.

\sin 30^\circ = 0.5, \quad \sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \sin 60^\circ = \frac{\sqrt{3}}{2}

Memorise the exact ratios for 30°, 45°, and 60° — they appear frequently.
Use exact angles when the ratio matches a known special value.
Round calculator results to one decimal place unless the problem asks for more precision.
Verify both angles by substituting back into the original equation.

CAST Rule and Quadrant Conversion Formulas

Quadrant	Angle Range	Positive Ratios	Formula for θ (given reference angle α)
I	0° to 90°	sin, cos, tan	θ = α
II	90° to 180°	sin only	θ = 180° − α
III	180° to 270°	tan only	θ = 180° + α
IV	270° to 360°	cos only	θ = 360° − α

Worked example

Example 1 — Finding Two Angles with a Given Sine Value

Find all angles θ in the interval 0° ≤ θ ≤ 360° such that sin θ = −0.82. Round to one decimal place.

Identify the ratio and its sign
The given ratio is $\sin\theta = -0.82$ . The value is negative. Sine represents the y-coordinate on the unit circle, so a negative sine means the terminal arm is below the x-axis.
$\sin\theta = -0.82$
Use CAST to find the correct quadrants
Sine is positive only in Quadrants I and II. Therefore sine is negative in Quadrants III and IV. Those are the two quadrants where the terminal arm must lie.
Find the reference angle
Ignore the negative sign and apply the inverse sine to the absolute value. Enter $\sin^{-1}(0.82)$ on a calculator to get the reference angle $\alpha$ .
$\alpha = \sin^{-1}(0.82) \approx 55.1°$
Convert the reference angle to Quadrant III
In Quadrant III, the angle formula is $\theta = 180^\circ + \alpha$ . Add the reference angle to 180°.
$\theta_1 = 180^\circ + 55.1^\circ = 235.1^\circ$
Convert the reference angle to Quadrant IV
In Quadrant IV, the angle formula is $\theta = 360^\circ - \alpha$ . Subtract the reference angle from 360°.
$\theta_2 = 360^\circ - 55.1^\circ = 304.9^\circ$
Verify both answers
Check each angle on a calculator. Entering $\sin(235.1^\circ)$ gives approximately $-0.82$ , and entering $\sin(304.9^\circ)$ also gives approximately $-0.82$ . Both match the given ratio, confirming both answers are correct.
$\sin(235.1^\circ) \approx -0.82, \quad \sin(304.9^\circ) \approx -0.82$

Answer: θ ≈ 235.1° and θ ≈ 304.9°

Check: Both angles lie in the correct quadrants (III and IV), both have reference angle 55.1°, and a calculator confirms sin(235.1°) ≈ −0.82 and sin(304.9°) ≈ −0.82.

Worked example

Example 2 — Finding Two Angles with a Given Cosine Value (Exact Special Value)

Find all angles θ in the interval 0° ≤ θ ≤ 360° such that cos θ = −(√3/2). Give exact answers.

Identify the ratio and its sign
The given ratio is $\cos\theta = -\frac{\sqrt{3}}{2}$ . The value is negative. Cosine represents the x-coordinate on the unit circle, so a negative cosine means the terminal arm is to the left of the y-axis.
$\cos\theta = -\frac{\sqrt{3}}{2}$
Use CAST to find the correct quadrants
Cosine is positive only in Quadrants I and IV. Therefore cosine is negative in Quadrants II and III. The terminal arm must be in one of those two quadrants.
Find the reference angle using the known special value
Recall that $\cos 60^\circ = \frac{\sqrt{3}}{2}$ , which matches the absolute value of the given ratio. Therefore the reference angle is exactly 60°. No calculator is needed here.
$\alpha = 60^\circ$
Convert the reference angle to Quadrant II
In Quadrant II, the angle formula is $\theta = 180^\circ - \alpha$ . Subtract the reference angle from 180°.
$\theta_1 = 180^\circ - 60^\circ = 120^\circ$
Convert the reference angle to Quadrant III
In Quadrant III, the angle formula is $\theta = 180^\circ + \alpha$ . Add the reference angle to 180°.
$\theta_2 = 180^\circ + 60^\circ = 240^\circ$
Verify both answers
Check each angle. The cosine of 120° equals $-\frac{\sqrt{3}}{2}$ , and the cosine of 240° also equals $-\frac{\sqrt{3}}{2}$ . Both results match the given ratio exactly.
$\cos(120^\circ) = -\frac{\sqrt{3}}{2}, \quad \cos(240^\circ) = -\frac{\sqrt{3}}{2}$

Answer: θ = 120° and θ = 240°

Check: Both angles have reference angle 60°, lie in Quadrants II and III respectively (where cosine is negative), and produce cos θ = −(√3/2) exactly.

Common mistakes and how to avoid them

Using the given ratio directly in the inverse function without removing the negative sign first, which produces a negative angle that is outside the 0°–360° range.

Correction: Always take the absolute value of the ratio before finding the reference angle. Apply the sign information separately using the CAST rule.

Choosing the wrong pair of quadrants — for example, selecting Quadrants I and II for a negative sine, when sine is negative in Quadrants III and IV.

Correction: Remember: a negative ratio means you want the quadrants where that ratio is NOT positive. CAST shows where each ratio is positive; the negative case is always the other two quadrants.

Applying the Quadrant II formula (180° − α) when the terminal arm is actually in Quadrant III, giving one correct and one wrong angle.

Correction: Write down both quadrant labels before calculating. Match each label to its specific formula: Q II → 180° − α, Q III → 180° + α, Q IV → 360° − α.

Forgetting to verify both answers by substituting back into the original ratio, and then missing a sign error.

Correction: Always substitute both angles into a calculator at the end. The displayed value must match the given ratio exactly (within rounding). If it does not, recheck the quadrant choice.

Rounding the reference angle too early and then using that rounded value in further calculations, causing both final angles to be slightly off.

Correction: Keep the full calculator value of the reference angle in memory and round only the final answer. State answers to one decimal place unless the problem specifies otherwise.

Lesson summary

A single trigonometric ratio value is shared by exactly two angles in the interval 0° to 360° (except at the boundary angles 0°, 90°, 180°, 270°, 360°).
The CAST rule identifies which two quadrants contain angles with the correct sign for a given ratio.
The reference angle α is found by applying the inverse trig function to the absolute value of the ratio.
Quadrant conversion formulas translate the reference angle into the actual angles: Q I: θ = α; Q II: θ = 180° − α; Q III: θ = 180° + α; Q IV: θ = 360° − α.
Special ratios like sin 30°, cos 60°, and tan 45° should be recognised and used as exact values without a calculator.
Always verify both angles by substituting them back into the original trigonometric expression on a calculator.

Check your understanding

Question 1

The value of cos θ is positive. According to the CAST rule, in which two quadrants could θ lie?

Quadrants I and II
Quadrants II and III
Quadrants I and IV
Quadrants III and IV

Show answer and explanation

Quadrants I and IV

The CAST rule shows that cosine is positive in Quadrant I (All positive) and Quadrant IV (Cosine positive). It is negative in Quadrants II and III.

Question 2

The reference angle for a trigonometric equation is α = 38°, and the terminal arm is in Quadrant III. What is θ?

38°
142°
218°
322°

Show answer and explanation

218°

In Quadrant III, the conversion formula is θ = 180° + α = 180° + 38° = 218°. Option 142° uses the Q II formula, and 322° uses the Q IV formula — both are wrong for Quadrant III.

Question 3

A student needs to solve sin θ = 0.5 for 0° ≤ θ ≤ 360°. What is the correct reference angle, and in which quadrants does the solution lie?

α = 30°, Quadrants I and II
α = 60°, Quadrants I and IV
α = 30°, Quadrants III and IV
α = 45°, Quadrants I and II

Show answer and explanation

α = 30°, Quadrants I and II

sin 30° = 0.5, so α = 30°. Because the ratio is positive, the CAST rule points to Quadrants I and II (where sine is positive). The two solutions are 30° and 150°.

Question 4

Which pair of angles both satisfy tan θ = −1 in the interval 0° ≤ θ ≤ 360°?

45° and 225°
135° and 225°
135° and 315°
45° and 315°

Show answer and explanation

135° and 315°

tan is negative in Quadrants II and IV. The reference angle is tan⁻¹(1) = 45°. Q II: 180° − 45° = 135°. Q IV: 360° − 45° = 315°. Verify: tan(135°) = −1 and tan(315°) = −1. Both correct.

Key terms

Terminal arm: The rotating ray that forms angle θ, starting from the positive x-axis and measured counterclockwise.
Reference angle: The positive acute angle (between 0° and 90°) between the terminal arm and the nearest part of the x-axis.
CAST rule: A memory tool stating which trigonometric ratio is positive in each quadrant: Cosine in Q IV, All in Q I, Sine in Q II, Tangent in Q III.
Unit circle: A circle with radius 1 centred at the origin; the point on the circle at angle θ has coordinates (cos θ, sin θ).
Quadrant: One of the four regions of the coordinate plane divided by the x-axis and y-axis, numbered I through IV counterclockwise from the top-right.
Inverse trigonometric function: A function such as sin⁻¹, cos⁻¹, or tan⁻¹ that returns the angle whose sine, cosine, or tangent equals a given value.
Special angles: The angles 30°, 45°, and 60° whose trigonometric ratios are exact values that should be memorised (e.g., sin 30° = 0.5).
Absolute value: The positive version of a number, ignoring its sign; written as |x|. Used here to find the reference angle before applying sign information from CAST.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation D1.3. It is a study resource, not an official curriculum publication.

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D1.3 · Find two angles with the same trigonometric ratio

Ontario Grade 11 Mathematics

Finding All Angles That Share the Same Sine, Cosine, or Tangent Value (MCR3U – D1.3)

What you will learn

Prerequisite Bridge: The Unit Circle and Quadrants

The CAST Rule: Which Ratios Are Positive Where?

Reference Angles: The Bridge Between Quadrants

Putting It All Together: A Four-Step Method

Special Exact Values and Boundary Cases

CAST Rule and Quadrant Conversion Formulas

Example 1 — Finding Two Angles with a Given Sine Value

Example 2 — Finding Two Angles with a Given Cosine Value (Exact Special Value)

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MCR3U

About this lesson