Trigonometry with Obtuse Angles Revision Notes for HSC SSCE Mathematics Standard

Trigonometry with Obtuse Angles

Understanding angle types

When working with trigonometry, it is essential to understand how the size of an angle affects the signs of trigonometric ratios. Angles are classified into two main categories based on their size:

Acute angles range from $0°$ to $90°$ . These are the angles you are most familiar with from basic right-angled triangle trigonometry.

Obtuse angles range from $90°$ to $180°$ . These angles are larger than a right angle but smaller than a straight line.

infoNote

The key difference between these angle types is how their trigonometric ratios behave, particularly whether the ratios are positive or negative values. Understanding this distinction is crucial for solving trigonometry problems correctly.

Signs of trigonometric ratios

The signs of the three main trigonometric ratios ( $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ ) depend on whether the angle is acute or obtuse. This is an important concept because your calculator automatically applies these rules when you calculate trigonometric values.

For acute angles (0° to 90°)

When dealing with acute angles, all three trigonometric ratios produce positive values:

$\sin \theta$ is positive
$\cos \theta$ is positive
$\tan \theta$ is positive

This makes sense from basic right-angled triangle trigonometry, where all side lengths are positive.

For obtuse angles (90° to 180°)

When working with obtuse angles, the situation changes significantly:

$\sin \theta$ is positive (stays positive)
$\cos \theta$ is negative (becomes negative)
$\tan \theta$ is negative (becomes negative)

infoNote

Your calculator automatically accounts for these sign changes when you enter an obtuse angle. You don't need to manually adjust the signs - the calculator handles this for you based on the angle size.

chatImportant

Exam tip: Remember that only sine stays positive for obtuse angles. Both cosine and tangent become negative. This is crucial for solving problems correctly and avoiding sign errors.

Finding trigonometric ratios of obtuse angles

When you need to find the value of a trigonometric ratio for an obtuse angle, your calculator handles the sign automatically. Simply enter the angle and the calculator will give you the correct value, including the appropriate sign.

lightbulbExample

Worked Example: Calculating trigonometric values

Question: Find the value of the following obtuse angles, correct to two decimal places:

a) $\sin 164.25°$

b) $\tan 124°30'$

Solution:

Part a:

Enter $\sin 164.25°$ into your calculator.

$\sin 164.25° = 0.2714404499$

$= 0.27 \text{ (rounded to two decimal places)}$

Notice that this value is positive, which is correct because sine is positive for obtuse angles.

Part b:

Enter $\tan 124°30'$ into your calculator. Note that $124°30'$ means $124$ degrees and $30$ minutes.

$\tan 124°30' = -1.455009029$

$= -1.46 \text{ (rounded to two decimal places)}$

Notice that this value is negative, which is correct because tangent is negative for obtuse angles.

Finding obtuse angles from trigonometric ratios

Sometimes you need to work backwards: given a trigonometric ratio, you need to find the angle. This involves using inverse trigonometric functions ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ ).

When you use inverse functions with negative ratios (for cosine or tangent), your calculator might return a negative angle. To find the corresponding obtuse angle, you need to add $180°$ to this negative result.

lightbulbExample

Worked Example: Finding angles from ratios

Question:

a) Given $\cos \theta = -0.5178$ , find the value of $\theta$ to the nearest degree.

b) Given $\tan \theta = -1.32$ , find the value of $\theta$ to the nearest minute.

Solution:

Part a:

Use the inverse cosine function on your calculator.

$\cos \theta = -0.5178$

Press SHIFT $\cos^{-1}(-0.5178)$ on your calculator.

$\theta = 121° \text{ (to the nearest degree)}$

The calculator gives the obtuse angle directly in this case.

Part b:

Use the inverse tangent function on your calculator.

$\tan \theta = -1.32$

Press SHIFT $\tan^{-1}(-1.32)$ on your calculator.

$\theta = -52.8533133°$

Since this is a negative angle, we need to find the corresponding obtuse angle. Add $180°$ to the negative result:

$\theta = 180° - 52.8533133°$

$= 127.1466867°$

Convert this to degrees and minutes:

$= 127°09' \text{ (to the nearest minute)}$

chatImportant

Key step: When your calculator gives you a negative angle for cosine or tangent, add $180°$ to find the obtuse angle in the range $90°$ to $180°$ .

bookmarkSummary

Key Points to Remember:

For acute angles (0° to 90°), all trigonometric ratios ( $\sin$ , $\cos$ , $\tan$ ) are positive.
For obtuse angles (90° to 180°), only $\sin \theta$ is positive, while $\cos \theta$ and $\tan \theta$ are both negative.
Your calculator automatically applies the correct signs when you calculate trigonometric ratios of obtuse angles.
When finding an obtuse angle from a negative cosine or tangent ratio, add 180° to the negative angle your calculator returns.
Always check your final answer makes sense: an obtuse angle must be between 90° and 180°.

Trigonometry with Obtuse Angles (HSC SSCE Mathematics Standard): Revision Notes