Sec 3 E-Math · Further trigonometry

Sine and cosine of obtuse angles questions with answers

Practice questions on sine and cosine of obtuse angles, 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

Given that $\sin 68^\circ = 0.927$ , write down the value of $\sin 112^\circ$ .

Step 1 of 1

Use the value you are given

\sin 112^\circ = \sin(180^\circ - 112^\circ)

\phantom{\sin 112^\circ} = \sin 68^\circ

\phantom{\sin 112^\circ} = 0.927

Try a question: Sine and cosine of obtuse angles

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Questions

Question 1sin(180° − x) = sin x
Given that $\sin 65^\circ = 0.906$ , write down the value of $\sin 115^\circ$ .
Show the worked solution
1. Step 1: Use the value you are given $\sin 115^\circ = \sin(180^\circ - 115^\circ)$ $\phantom{\sin 115^\circ} = \sin 65^\circ$ $\phantom{\sin 115^\circ} = 0.906$
Question 2cos(180° − x) = −cos x
Given that $\cos 83^\circ = 0.122$ , write down the value of $\cos 97^\circ$ .
Show the worked solution
1. Step 1: Use the value you are given $\cos 97^\circ = -\cos(180^\circ - 97^\circ)$ $\phantom{\cos 97^\circ} = -\cos 83^\circ$ $\phantom{\cos 97^\circ} = -0.122$
Question 3Find the obtuse angle
Given that $\cos x = -0.8$ and $0^\circ < x < 180^\circ$ , find $x$ , correct to 1 decimal place.
Show the worked solution
1. Step 1: Use the inverse ratio on the calculator $\cos x = -0.8$ $x = \cos^{-1}(-0.8)$ $x = 143.1301\ldots^\circ$ $\phantom{x} = 143.1^\circ$ (1 d.p.)
2. Step 2: Check with cos x = −cos(180° − x) $\cos 143.1^\circ = -\cos(180^\circ - 143.1^\circ)$ $\phantom{\cos 143.1^\circ} = -\cos 36.9^\circ \approx -0.8$ ✓
Question 4Both angles with the same sine
Given that $\sin x = 0.88$ and $0^\circ < x < 180^\circ$ , find the two possible values of $x$ , correct to 1 decimal place.
Show the worked solution
1. Step 1: The calculator gives the acute angle $\sin x = 0.88$ $x = \sin^{-1}(0.88)$ $x = 61.6423\ldots^\circ$ $\phantom{x} = 61.6^\circ$ (1 d.p.)
2. Step 2: Use sin x = sin(180° − x) for the obtuse one $\sin x = \sin(180^\circ - 61.6423\ldots^\circ)$ $x = 180^\circ - 61.6423\ldots^\circ$ $\phantom{x} = 118.4^\circ$ (1 d.p.)
3. Step 3: Write both answers $x = 61.6^\circ$ or $x = 118.4^\circ$
Question 5Exact values for an obtuse angle
Given that $\cos \theta = -\dfrac{5}{13}$ and that $\theta$ is obtuse, find, without using a calculator, the value of $\sin \theta$ .
Show the worked solution
1. Step 1: The basic angle α = 180° − θ: a right-angled triangle with the ratio's two sidesAdjacent $= 5$ , hypotenuse $= 13$ : opposite $= \sqrt{13^2 - 5^2}$ Opposite $= \sqrt{144}$ $= 12$ (Pythagoras' theorem)
2. Step 2: θ is obtuse: sin θ is positive, cos θ is negative $\sin \theta = \sin \alpha$ ( $\theta$ obtuse) $\sin \theta$ $= \dfrac{12}{13}$

More on further trigonometry

Notes: Further trigonometry Worksheet with answers (PDF)

Endless questions on sine and cosine of obtuse angles

A free account gives you new questions every time, step-by-step help, and tracks what you've mastered.

Start free

Worked example

Given that

\sin 68^\circ = 0.927

, write down the value of

\sin 112^\circ

Step 1 of 1

Use the value you are given

\sin 112^\circ = \sin(180^\circ - 112^\circ)

\phantom{\sin 112^\circ} = \sin 68^\circ

\phantom{\sin 112^\circ} = 0.927

Questions

Question 1sin(180° − x) = sin x

Given that $\sin 65^\circ = 0.906$ , write down the value of $\sin 115^\circ$ .

Show the worked solution

Step 1: Use the value you are given $\sin 115^\circ = \sin(180^\circ - 115^\circ)$ $\phantom{\sin 115^\circ} = \sin 65^\circ$ $\phantom{\sin 115^\circ} = 0.906$

Question 2cos(180° − x) = −cos x

Given that $\cos 83^\circ = 0.122$ , write down the value of $\cos 97^\circ$ .

Show the worked solution

Step 1: Use the value you are given $\cos 97^\circ = -\cos(180^\circ - 97^\circ)$ $\phantom{\cos 97^\circ} = -\cos 83^\circ$ $\phantom{\cos 97^\circ} = -0.122$

Question 3Find the obtuse angle

Given that $\cos x = -0.8$ and $0^\circ < x < 180^\circ$ , find $x$ , correct to 1 decimal place.

Show the worked solution

Step 1: Use the inverse ratio on the calculator $\cos x = -0.8$ $x = \cos^{-1}(-0.8)$ $x = 143.1301\ldots^\circ$ $\phantom{x} = 143.1^\circ$ (1 d.p.)
Step 2: Check with cos x = −cos(180° − x) $\cos 143.1^\circ = -\cos(180^\circ - 143.1^\circ)$ $\phantom{\cos 143.1^\circ} = -\cos 36.9^\circ \approx -0.8$ ✓

Question 4Both angles with the same sine

Given that $\sin x = 0.88$ and $0^\circ < x < 180^\circ$ , find the two possible values of $x$ , correct to 1 decimal place.

Show the worked solution

Step 1: The calculator gives the acute angle $\sin x = 0.88$ $x = \sin^{-1}(0.88)$ $x = 61.6423\ldots^\circ$ $\phantom{x} = 61.6^\circ$ (1 d.p.)
Step 2: Use sin x = sin(180° − x) for the obtuse one $\sin x = \sin(180^\circ - 61.6423\ldots^\circ)$ $x = 180^\circ - 61.6423\ldots^\circ$ $\phantom{x} = 118.4^\circ$ (1 d.p.)
Step 3: Write both answers $x = 61.6^\circ$ or $x = 118.4^\circ$

Question 5Exact values for an obtuse angle

Given that $\cos \theta = -\dfrac{5}{13}$ and that $\theta$ is obtuse, find, without using a calculator, the value of $\sin \theta$ .

Show the worked solution

Step 1: The basic angle α = 180° − θ: a right-angled triangle with the ratio's two sidesAdjacent $= 5$ , hypotenuse $= 13$ : opposite $= \sqrt{13^2 - 5^2}$ Opposite $= \sqrt{144}$ $= 12$ (Pythagoras' theorem)
Step 2: θ is obtuse: sin θ is positive, cos θ is negative $\sin \theta = \sin \alpha$ ( $\theta$ obtuse) $\sin \theta$ $= \dfrac{12}{13}$