Sec 3 A-Math · Trigonometric equations

Principal values and the basic angle questions with answers

Practice questions on principal values and the basic angle, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Use your calculator to find the principal value of $\cos^{-1}\left(-\dfrac{1}{2}\right)$ . Hence write down the basic angle.

Step 1 of 3

Calculator (in degrees): this is the principal value

\cos^{-1}\left(-\dfrac{1}{2}\right)

= 120^\circ

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Questions

Question 1Exact values: the calculator's angle, then the basic angle
Use your calculator to find the principal value of $\tan^{-1}\left(-1\right)$ . Hence write down the basic angle.
Show the worked solution
1. Step 1: Calculator (in degrees): this is the principal value $\tan^{-1}\left(-1\right) = -45^\circ$
2. Step 2: Which quadrant is the calculator's angle in?It is in the 4th quadrant (a negative angle, measured clockwise).
3. Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = -(-45^\circ)$ $\alpha = 45^\circ$ Check: $\tan^{-1}\left(1\right) = 45^\circ$ , the positive value gives the basic angle straight away.
Question 2Exact values: the calculator's angle, then the basic angle
Use your calculator to find the principal value of $\tan^{-1}\left(-\sqrt{3}\right)$ . Hence write down the basic angle.
Show the worked solution
1. Step 1: Calculator (in degrees): this is the principal value $\tan^{-1}\left(-\sqrt{3}\right) = -60^\circ$
2. Step 2: Which quadrant is the calculator's angle in?It is in the 4th quadrant (a negative angle, measured clockwise).
3. Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = -(-60^\circ)$ $\alpha = 60^\circ$ Check: $\tan^{-1}\left(\sqrt{3}\right) = 60^\circ$ , the positive value gives the basic angle straight away.
Question 3Decimal values, to 1 decimal place
Use your calculator to find the principal value of $\tan^{-1}\left(-0.3\right)$ . Hence write down the basic angle, correct to 1 decimal place.
Show the worked solution
1. Step 1: Calculator (in degrees): this is the principal value, to 1 decimal place $\tan^{-1}\left(-0.3\right) = -16.7^\circ$
2. Step 2: Which quadrant is the calculator's angle in?It is in the 4th quadrant (a negative angle, measured clockwise).
3. Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = -(-16.7^\circ)$ $\alpha = 16.7^\circ$ Check: $\tan^{-1}\left(0.3\right) = 16.7^\circ$ , the positive value gives the basic angle straight away.
Question 4Decimal values, to 1 decimal place
Use your calculator to find the principal value of $\tan^{-1}\left(0.6\right)$ . Hence write down the basic angle, correct to 1 decimal place.
Show the worked solution
1. Step 1: Calculator (in degrees): this is the principal value, to 1 decimal place $\tan^{-1}\left(0.6\right) = 31^\circ$
2. Step 2: Which quadrant is the calculator's angle in?It is in the 1st quadrant.
3. Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = 31^\circ$ $\alpha = 31^\circ$ Check: $\tan^{-1}\left(0.6\right) = 31^\circ$ , the positive value gives the basic angle straight away.

More on trigonometric equations

Notes: Trigonometric equations Worksheet with answers (PDF)

Endless questions on principal values and the basic angle

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Worked example

Use your calculator to find the principal value of

\cos^{-1}\left(-\dfrac{1}{2}\right)

. Hence write down the basic angle.

Step 1 of 3

Calculator (in degrees): this is the principal value

\cos^{-1}\left(-\dfrac{1}{2}\right)

= 120^\circ

Browsing

Account

Read the notes and print worksheets and papers

✓

Try a question with marking, hints and the step guide

✓

Endless questions on every chapter, at every level

–

✓

Levels that unlock as you improve

–

✓

Your mistakes tracked by type, so you see what keeps costing marks

–

✓

WA practice papers on the chapters being tested

–

✓

Revision reminders before a topic fades

–

✓

Progress page and leaderboards

–

✓

Questions

Question 1Exact values: the calculator's angle, then the basic angle

Use your calculator to find the principal value of $\tan^{-1}\left(-1\right)$ . Hence write down the basic angle.

Show the worked solution

Step 1: Calculator (in degrees): this is the principal value $\tan^{-1}\left(-1\right) = -45^\circ$
Step 2: Which quadrant is the calculator's angle in?It is in the 4th quadrant (a negative angle, measured clockwise).
Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = -(-45^\circ)$ $\alpha = 45^\circ$ Check: $\tan^{-1}\left(1\right) = 45^\circ$ , the positive value gives the basic angle straight away.

Question 2Exact values: the calculator's angle, then the basic angle

Use your calculator to find the principal value of $\tan^{-1}\left(-\sqrt{3}\right)$ . Hence write down the basic angle.

Show the worked solution

Step 1: Calculator (in degrees): this is the principal value $\tan^{-1}\left(-\sqrt{3}\right) = -60^\circ$
Step 2: Which quadrant is the calculator's angle in?It is in the 4th quadrant (a negative angle, measured clockwise).
Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = -(-60^\circ)$ $\alpha = 60^\circ$ Check: $\tan^{-1}\left(\sqrt{3}\right) = 60^\circ$ , the positive value gives the basic angle straight away.

Question 3Decimal values, to 1 decimal place

Use your calculator to find the principal value of $\tan^{-1}\left(-0.3\right)$ . Hence write down the basic angle, correct to 1 decimal place.

Show the worked solution

Step 1: Calculator (in degrees): this is the principal value, to 1 decimal place $\tan^{-1}\left(-0.3\right) = -16.7^\circ$
Step 2: Which quadrant is the calculator's angle in?It is in the 4th quadrant (a negative angle, measured clockwise).
Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = -(-16.7^\circ)$ $\alpha = 16.7^\circ$ Check: $\tan^{-1}\left(0.3\right) = 16.7^\circ$ , the positive value gives the basic angle straight away.

Question 4Decimal values, to 1 decimal place

Use your calculator to find the principal value of $\tan^{-1}\left(0.6\right)$ . Hence write down the basic angle, correct to 1 decimal place.

Show the worked solution

Step 1: Calculator (in degrees): this is the principal value, to 1 decimal place $\tan^{-1}\left(0.6\right) = 31^\circ$
Step 2: Which quadrant is the calculator's angle in?It is in the 1st quadrant.
Step 3: The basic angle α: the acute angle between the arm and the x-axis (always positive) $\alpha = 31^\circ$ $\alpha = 31^\circ$ Check: $\tan^{-1}\left(0.6\right) = 31^\circ$ , the positive value gives the basic angle straight away.