Sine, cosine and tangent up to 180 degrees

Q: How is an obtuse angle related to its supplementary reference angle in trigonometry?

Every obtuse angle ̂\(\theta\) (between \(90^\circ\) and \(180^\circ\)) has a supplementary acute reference angle given by \(180^\circ - \theta\).

Q: What is the rule relating the sine of an obtuse angle to the sine of its acute supplementary angle?

\(\sin(180^\circ - \theta) = \sin\theta\), meaning sine remains positive for all angles strictly between \(0^\circ\) and \(180^\circ\).

Q: How do the signs and values of cosine and tangent change when an angle becomes obtuse?

Both cosine and tangent become negative: \(\cos(180^\circ - \theta) = -\cos\theta\) and \(\tan(180^\circ - \theta) = -\tan\theta\).

Q: What are the exact values of sine, cosine, and tangent at the boundary angles \(0^\circ\), \(90^\circ\), and \(180^\circ\)?

\(\sin 0^\circ = 0\), \(\cos 0^\circ = 1\), \(\tan 0^\circ = 0\); \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\tan 90^\circ\) is undefined; \(\sin 180^\circ = 0\), \(\cos 180^\circ = -1\), \(\tan 180^\circ = 0\).

Q: Why does the equation \(\sin\theta = k\) (where \(0?

Because sine is positive in both the acute and obtuse regions: the first solution is \(\theta_1 = \sin^{-1}(k)\) and the second is \(\theta_2 = 180^\circ - \theta_1\).

Q: How do you solve \(\cos\theta = k\) for \(0^\circ \le \theta \le 180^\circ\) when \(k\) is negative?

Evaluate \(\theta = \cos^{-1}(k)\) directly on your calculator, or find the acute reference angle \(\cos^{-1}(|k|)\) and subtract it from \(180^\circ\).

Q: How do you find the angle \(\theta\) between \(0^\circ\) and \(180^\circ\) when solving \(\tan\theta = k\) for \(k?

Add \(180^\circ\) to the negative acute value given by your calculator: \(\theta = 180^\circ + \tan^{-1}(k)\).

Q: What calculator angle mode must be used for Pearson Edexcel Specification B trigonometry problems?

The calculator must be set to Degree mode (indicated by D or DEG on the screen).

Practice

Mathematics (Specification B) · Pearson Edexcel IGCSE · 7 min read · Updated: 22 Oct 2026

Introduction: Beyond Right-Angled Triangles

Welcome to trigonometry beyond the right-angled triangle! In earlier studies, you used sine, cosine, and tangent inside right-angled triangles using the familiar memory aid SOH CAH TOA. But what happens when an angle is greater than \(90^\circ\)? Can we find the sine of \(120^\circ\) or the cosine of \(150^\circ\)?

The answer is a definite yes! In Pearson Edexcel International GCSE Mathematics (Specification B), you are required to understand and calculate the trigonometric ratios of angles up to \(180^\circ\). This is an essential skill, particularly when working with obtuse angles in later topics such as the sine rule, cosine rule, and the area of a triangle formula \(\text{Area} = \frac{1}{2}ab\sin C\).

Don't worry if this feels unfamiliar at first. By understanding a few simple symmetry rules and learning how your calculator handles these values, you will master these questions easily.

Angles up to \(180^\circ\): Acute, Right, and Obtuse

Let us classify the angles we meet in this chapter:

1. Acute angles: Angles strictly between \(0^\circ\) and \(90^\circ\) (\(0^\circ < \theta < 90^\circ\)).
2. Right angle: An angle of exactly \(90^\circ\) (\(\theta = 90^\circ\)).
3. Obtuse angles: Angles strictly between \(90^\circ\) and \(180^\circ\) (\(90^\circ < \theta < 180^\circ\)).
4. Boundary angles: The specific values \(\theta = 0^\circ\), \(\theta = 90^\circ\), and \(\theta = 180^\circ\).

The Supplementary Angle Rule (\(180^\circ - \theta\))

An obtuse angle \(\theta\) has a matching acute angle known as its supplementary reference angle, given by \((180^\circ - \theta)\). Because angles on a straight line add up to \(180^\circ\), every obtuse angle shares a fundamental geometric link with its supplementary acute partner.

Here are the three golden rules you must know for any angle \(\theta\) between \(0^\circ\) and \(180^\circ\):

Rule 1: Sine is always positive for angles up to \(180^\circ\)
\(\sin(180^\circ - \theta) = \sin\theta\)
Example: \(\sin 150^\circ = \sin(180^\circ - 150^\circ) = \sin 30^\circ = 0.5\)
Example: \(\sin 135^\circ = \sin(180^\circ - 135^\circ) = \sin 45^\circ \approx 0.7071\)

Rule 2: Cosine of an obtuse angle is negative
\(\cos(180^\circ - \theta) = -\cos\theta\)
Example: \(\cos 120^\circ = -\cos(180^\circ - 120^\circ) = -\cos 60^\circ = -0.5\)
Example: \(\cos 150^\circ = -\cos 30^\circ \approx -0.8660\)

Rule 3: Tangent of an obtuse angle is negative
\(\tan(180^\circ - \theta) = -\tan\theta\)
Example: \(\tan 135^\circ = -\tan(180^\circ - 135^\circ) = -\tan 45^\circ = -1\)
Example: \(\tan 120^\circ = -\tan 60^\circ \approx -1.732\)

Summary of Signs between \(0^\circ\) and \(180^\circ\)

For an acute angle (\(0^\circ < \theta < 90^\circ\)):
\(\sin\theta > 0\) (positive)
\(\cos\theta > 0\) (positive)
\(\tan\theta > 0\) (positive)

For an obtuse angle (\(90^\circ < \theta < 180^\circ\)):
\(\sin\theta > 0\) (positive)
\(\cos\theta < 0\) (negative)
\(\tan\theta < 0\) (negative)

Key Takeaway: Between \(0^\circ\) and \(180^\circ\), sine is always positive, while cosine and tangent turn negative once the angle becomes obtuse (greater than \(90^\circ\)).

Trigonometric Values at the Boundaries

You should be familiar with the values at the key points \(0^\circ\), \(90^\circ\), and \(180^\circ\):

At \(\theta = 0^\circ\):
\(\sin 0^\circ = 0\)
\(\cos 0^\circ = 1\)
\(\tan 0^\circ = 0\)

At \(\theta = 90^\circ\):
\(\sin 90^\circ = 1\)
\(\cos 90^\circ = 0\)
\(\tan 90^\circ\) is undefined (or gives a Math Error on your calculator, because division by zero occurs as \(\cos 90^\circ = 0\)).

At \(\theta = 180^\circ\):
\(\sin 180^\circ = 0\)
\(\cos 180^\circ = -1\)
\(\tan 180^\circ = 0\)

Solving Simple Trigonometric Equations for \(0^\circ \le \theta \le 180^\circ\)

In examination questions, you are often given a trigonometric value and asked to find the possible angle(s) \(\theta\) in the range \(0^\circ \le \theta \le 180^\circ\).

1. Solving \(\sin\theta = k\) (The Two-Solution Case)

Because \(\sin\theta\) is positive in both the acute and obtuse ranges, when \(0 < k < 1\), there are two possible angles between \(0^\circ\) and \(180^\circ\):

Step 1: Find the acute angle using your calculator: \(\theta_1 = \sin^{-1}(k)\).
Step 2: Find the second (obtuse) angle by subtracting from \(180^\circ\): \(\theta_2 = 180^\circ - \theta_1\).

Worked Example: Solve \(\sin\theta = 0.5\) for \(0^\circ \le \theta \le 180^\circ\).
\(\theta_1 = \sin^{-1}(0.5) = 30^\circ\)
\(\theta_2 = 180^\circ - 30^\circ = 150^\circ\)
Therefore, the solutions are \(\theta = 30^\circ\) and \(\theta = 150^\circ\).

Note on special cases: If \(\sin\theta = 1\), there is only one solution: \(\theta = 90^\circ\). If \(\sin\theta = 0\), the solutions in the interval are \(\theta = 0^\circ\) and \(\theta = 180^\circ\).

2. Solving \(\cos\theta = k\) (One-to-One Case)

Unlike sine, cosine has a unique value for every angle between \(0^\circ\) and \(180^\circ\):

Case A: If \(k > 0\) (positive value):
The angle is acute (\(0^\circ < \theta < 90^\circ\)).
Calculate directly: \(\theta = \cos^{-1}(k)\).
Example: If \(\cos\theta = 0.8\), \(\theta = \cos^{-1}(0.8) \approx 36.87^\circ\).

Case B: If \(k < 0\) (negative value):
The angle is obtuse (\(90^\circ < \theta < 180^\circ\)).
Your calculator automatically gives the obtuse angle when you enter the negative value!
Example: If \(\cos\theta = -0.5\), typing \(\cos^{-1}(-0.5)\) into your scientific calculator yields \(\theta = 120^\circ\).
Alternatively, use the acute reference angle: \(\cos^{-1}(0.5) = 60^\circ\), so \(\theta = 180^\circ - 60^\circ = 120^\circ\).

3. Solving \(\tan\theta = k\)

Tangent also yields a single angle within this range:

If \(k > 0\): \(\theta\) is acute. \(\theta = \tan^{-1}(k)\).
Example: If \(\tan\theta = 1\), \(\theta = \tan^{-1}(1) = 45^\circ\).

If \(k < 0\): \(\theta\) is obtuse.
If you type \(\tan^{-1}(-1)\) on a calculator, it gives \(-45^\circ\) (because standard calculators return values between \(-90^\circ\) and \(90^\circ\)). To obtain the angle between \(0^\circ\) and \(180^\circ\), simply add \(180^\circ\):
\(\theta = 180^\circ + (-45^\circ) = 135^\circ\).
Alternatively, calculate the positive reference angle first: \(\tan^{-1}(1) = 45^\circ\), then calculate \(\theta = 180^\circ - 45^\circ = 135^\circ\).

Step-by-Step Practical Examples

Example 1: Determining the Type of Angle

An angle \(x\) lies between \(0^\circ\) and \(180^\circ\). Given that \(\cos x = -0.4226\):
(a) State whether \(x\) is acute or obtuse.
(b) Calculate the value of \(x\) to 1 decimal place.

Solution:
(a) Since the cosine of \(x\) is negative, \(x\) must be an obtuse angle (\(90^\circ < x < 180^\circ\)).
(b) Using a calculator: \(x = \cos^{-1}(-0.4226) \approx 115.0^\circ\).

Example 2: Two Solutions for Sine

Solve the equation \(4\sin\theta - 3 = 0\) for \(0^\circ \le \theta \le 180^\circ\). Give your answers to 1 decimal place.

Solution:
Step 1: Rearrange the equation to isolate \(\sin\theta\):
\(4\sin\theta = 3 \implies \sin\theta = \frac{3}{4} = 0.75\)
Step 2: Find the primary acute solution using \(\sin^{-1}\):
\(\theta_1 = \sin^{-1}(0.75) \approx 48.6^\circ\)
Step 3: Find the supplementary obtuse solution:
\(\theta_2 = 180^\circ - 48.59037...^\circ \approx 131.4^\circ\)
Answers: \(\theta = 48.6^\circ\) or \(\theta = 131.4^\circ\).

Example 3: Using Supplementary Angles Directly

Given that \(\sin 25^\circ \approx 0.4226\) and \(\cos 25^\circ \approx 0.9063\), state the exact decimal values of:
(a) \(\sin 155^\circ\)
(b) \(\cos 155^\circ\)

Solution:
Notice that \(155^\circ = 180^\circ - 25^\circ\).
(a) \(\sin 155^\circ = \sin(180^\circ - 25^\circ) = \sin 25^\circ = 0.4226\)
(b) \(\cos 155^\circ = -\cos(180^\circ - 25^\circ) = -\cos 25^\circ = -0.9063\)

Essential Calculator Advice and Conventions

1. Check Your Angle Mode: The Pearson Edexcel Specification B requires angles in degrees and decimals of a degree (radian measure is excluded from this syllabus). Make sure your calculator displays a small D or DEG at the top of the screen. If it shows R (radians) or G (gradians), your calculations will be wrong!

2. Degrees and Decimals of a Degree: Exam questions typically specify angles to 1 decimal place or significant figures. Unless told otherwise, calculate with full precision in your calculator's memory and round only at the final step.

3. Inverse Functions: To access \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\), press the SHIFT or 2ndF button followed by the \(\sin\), \(\cos\), or \(\tan\) key.

Common Mistakes to Avoid

Mistake 1: Forgetting the second solution when solving for sine
If a question asks for all values of \(\theta\) in \(0^\circ \le \theta \le 180^\circ\) satisfying \(\sin\theta = k\) (where \(0 < k < 1\)), students often provide only the acute angle from their calculator. Always remember: the second solution is \(180^\circ - \theta\).

Mistake 2: Missing the minus sign on the cosine of an obtuse angle
Remember that for an obtuse angle, \(\cos\theta\) is negative. If you construct a triangle calculation in geometry and obtain a negative cosine, that simply means the angle is greater than \(90^\circ\).

Mistake 3: Typing negative angles for tangent
When you evaluate \(\tan^{-1}\) of a negative number on a calculator, it produces a negative acute angle (e.g., \(-30^\circ\)). For angles in the range \(0^\circ\) to \(180^\circ\), you must add \(180^\circ\) to find the valid obtuse angle: \(180^\circ + (-30^\circ) = 150^\circ\).

Quick Revision Checklist

Can you answer these quickly before moving on?
1. If \(\sin\theta = 0.6\), what are the two values of \(\theta\) between \(0^\circ\) and \(180^\circ\)? (Answer: \(36.9^\circ\) and \(143.1^\circ\))
2. What is \(\cos 180^\circ\)? (Answer: \(-1\))
3. Is \(\cos 110^\circ\) positive or negative? (Answer: Negative)
4. Is \(\sin 110^\circ\) positive or negative? (Answer: Positive)
5. Why is \(\tan 90^\circ\) undefined? (Answer: Because \(\cos 90^\circ = 0\), and division by zero is undefined)

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

How is an obtuse angle related to its supplementary reference angle in trigonometry?

Every obtuse angle ̂\(\theta\) (between \(90^\circ\) and \(180^\circ\)) has a supplementary acute reference angle given by \(180^\circ - \theta\).

What is the rule relating the sine of an obtuse angle to the sine of its acute supplementary angle?

\(\sin(180^\circ - \theta) = \sin\theta\), meaning sine remains positive for all angles strictly between \(0^\circ\) and \(180^\circ\).

How do the signs and values of cosine and tangent change when an angle becomes obtuse?

Both cosine and tangent become negative: \(\cos(180^\circ - \theta) = -\cos\theta\) and \(\tan(180^\circ - \theta) = -\tan\theta\).

What are the exact values of sine, cosine, and tangent at the boundary angles \(0^\circ\), \(90^\circ\), and \(180^\circ\)?

\(\sin 0^\circ = 0\), \(\cos 0^\circ = 1\), \(\tan 0^\circ = 0\); \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\tan 90^\circ\) is undefined; \(\sin 180^\circ = 0\), \(\cos 180^\circ = -1\), \(\tan 180^\circ = 0\).

Why does the equation \(\sin\theta = k\) (where \(0 < k < 1\)) have two solutions in the interval \(0^\circ \le \theta \le 180^\circ\), and how are they found?

Because sine is positive in both the acute and obtuse regions: the first solution is \(\theta_1 = \sin^{-1}(k)\) and the second is \(\theta_2 = 180^\circ - \theta_1\).

How do you solve \(\cos\theta = k\) for \(0^\circ \le \theta \le 180^\circ\) when \(k\) is negative?

Evaluate \(\theta = \cos^{-1}(k)\) directly on your calculator, or find the acute reference angle \(\cos^{-1}(|k|)\) and subtract it from \(180^\circ\).

How do you find the angle \(\theta\) between \(0^\circ\) and \(180^\circ\) when solving \(\tan\theta = k\) for \(k < 0\)?

Add \(180^\circ\) to the negative acute value given by your calculator: \(\theta = 180^\circ + \tan^{-1}(k)\).

What calculator angle mode must be used for Pearson Edexcel Specification B trigonometry problems?

The calculator must be set to Degree mode (indicated by D or DEG on the screen).

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