Ratios of Angles Greater than 90° Revision Notes for Leaving Cert Mathematics

Ratios of Angles Greater than 90°

The unit circle

The unit circle is a circle with its centre at the origin (0, 0) and a radius of exactly 1 unit. This special circle becomes our foundation for understanding trigonometric ratios beyond the basic right triangle.

When we place any point P on the unit circle, something remarkable happens. The coordinates of this point are directly related to the trigonometric ratios of the angle θ (theta) that the radius makes with the positive x-axis.

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Key relationship: The coordinates of any point on the unit circle are P(cos θ, sin θ).

This means that for any angle θ, we can read the values of cos θ and sin θ directly from the x and y coordinates of the corresponding point on the unit circle.

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The unit circle intersects the coordinate axes at four important points:

(1, 0) at 0° and 360°
(0, 1) at 90°
(-1, 0) at 180°
(0, -1) at 270°

From these special points, we can determine exact values. For example, the point (-1, 0) gives us $\cos 180° = -1$ and $\sin 180° = 0$ .

The four quadrants

The coordinate axes divide a complete rotation of 360° into four equal sections called quadrants. Each quadrant has specific characteristics that determine whether trigonometric ratios are positive or negative.

Understanding which quadrant an angle lies in becomes crucial when working with angles greater than 90°, as this determines the sign of our trigonometric ratios.

The CAST rule

The CAST rule is an essential memory tool that tells us which trigonometric ratios are positive in each quadrant. The letters S, A, T, C represent which ratios are positive in each quadrant, starting from the second quadrant and moving clockwise.

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CAST rule breakdown:

Second quadrant (S): Only Sine is positive
First quadrant (A): All ratios are positive
Fourth quadrant (C): Only Cosine is positive
Third quadrant (T): Only Tangent is positive

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Memory tip: Remember "All Students Take Calculus" going anti-clockwise from the first quadrant.

Finding ratios for angles between 90° and 360°

When we need to find the exact value of a trigonometric ratio for an angle greater than 90°, we follow a systematic four-step approach:

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Four-Step Method:

Step 1: Make a rough sketch to determine which quadrant the angle lies in.

Step 2: Use the CAST rule to determine whether the ratio will be positive or negative.

Step 3: Find the reference angle - this is the acute angle (less than 90°) between the terminal side of the angle and the x-axis.

Step 4: Use the reference angle with special triangles (30°, 45°, 60°) or a calculator to find the numerical value, then apply the correct sign from Step 2.

Reference angles

The reference angle is the acute angle between the terminal side of the given angle and the x-axis. This concept is crucial for finding exact values using our knowledge of special triangles.

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Reference angle formulas for different quadrants:

Second quadrant: Reference angle = 180° - given angle
Third quadrant: Reference angle = given angle - 180°
Fourth quadrant: Reference angle = 360° - given angle

Special triangles

Two special right triangles are particularly useful for finding exact values and appear frequently in examinations:

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45-45-90 triangle:

Sides in ratio 1 : 1 : $\sqrt{2}$
$\cos 45° = \sin 45° = \frac{1}{\sqrt{2}}$

30-60-90 triangle:

Sides in ratio 1 : $\sqrt{3}$ : 2
$\sin 30° = \cos 60° = \frac{1}{2}$
$\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}$

Worked examples

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Worked Example 1: Finding exact values in surd form

Find: (i) $\sin 120°$ and (ii) $\cos 225°$

(i) $\sin 120°$ :

120° is in the second quadrant
Using CAST rule: sine is positive in the second quadrant
Reference angle = 180° - 120° = 60°
From the 30-60-90 triangle: $\sin 60° = \frac{\sqrt{3}}{2}$
Therefore: $\sin 120° = +\frac{\sqrt{3}}{2}$

(ii) $\cos 225°$ :

225° is in the third quadrant
Using CAST rule: cosine is negative in the third quadrant
Reference angle = 225° - 180° = 45°
From the 45-45-90 triangle: $\cos 45° = \frac{1}{\sqrt{2}}$
Therefore: $\cos 225° = -\frac{1}{\sqrt{2}}$

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Worked Example 2: Finding angles from given ratio values

Given: $\cos A = -0.4153$ and $0° ≤ A ≤ 360°$

Step 1: Since cosine is negative, A must be in the second or third quadrant (where cosine is negative according to CAST rule).

Step 2: Find the reference angle by using the inverse cosine function on the positive value: Reference angle = $\cos^{-1}(0.4153) = 65.5°$ (to 1 decimal place)

Step 3: Find the two possible values:

Second quadrant: $A = 180° - 65.5° = 114.5°$
Third quadrant: $A = 180° + 65.5° = 245.5°$

Therefore: $A = 114.5°$ and $245.5°$

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Exam tips:

Always sketch the angle to visualise which quadrant it's in
Use CAST rule consistently to determine the sign
Remember the special triangles - they appear frequently in exams
Check your calculator is in degree mode when finding reference angles
Be careful with negative signs - they're easy to drop or add incorrectly

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Key Points to Remember:

The unit circle shows that coordinates of any point are (cos θ, sin θ)
CAST rule determines signs: All positive in 1st, Sine in 2nd, Tangent in 3rd, Cosine in 4th quadrant
Reference angles are always acute and help us use special triangles for exact values
The four-step method works for any angle: sketch, determine sign, find reference angle, calculate
Special triangles (30-60-90 and 45-45-90) give exact values for common angles

Ratios of Angles Greater than 90° (Leaving Cert Mathematics): Revision Notes