C7.2; E7.2 - Finding angles with trigonometry

C7.2; E7.2 - Finding angles with trigonometry

A right triangle has perpendicular sides 6 cm and 11 cm. How can you find its acute angles without measuring a drawing? C7.2 and E7.2 use inverse trigonometric functions to recover angles from side ratios and combine these with Pythagoras in two-dimensional problems.

Inverse ratios

Sine, cosine and tangent turn an angle into a ratio. An inverse trigonometric function reverses that direction: it turns a ratio into an angle. First identify the known sides relative to the requested angle, just as when finding a side.

Finding an acute angle

In a right-angled triangle,

θ=sin⁡−1(oppositehypotenuse)θ=cos⁡−1(adjacenthypotenuse)θ=tan⁡−1(oppositeadjacent)\begin{aligned} \theta&=\sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\\ \theta&=\cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\\ \theta&=\tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) \end{aligned}

Use the function matching the known side pair, in degree mode. Not on the supplied formula list for Papers 1–4: learn how to reverse each ratio.

The notation sin⁡−1\sin^{-1} here means inverse sine, often labelled “asin”. It does not mean 1/sin⁡θ1/\sin\theta. On many calculators the inverse function is accessed using a second-function key; check that the whole fraction lies inside its brackets.

Worked example 1: an angle from two shorter sides

Question. Relative to acute angle θ\theta, a right triangle has opposite side 5 cm and adjacent side 12 cm. Calculate θ\theta.

Plan. Use tangent for opposite and adjacent, then reverse it.

tan⁡θ=512θ=tan⁡−1(512)=22.619864…∘\begin{aligned} \tan\theta&=\frac{5}{12}\\ \theta&=\tan^{-1}\left(\frac{5}{12}\right)=22.619864\ldots^\circ \end{aligned}

Answer. 22.6∘22.6^\circ to one decimal place.

Check. In degree mode, tan⁡22.619864…∘=5/12\tan22.619864\ldots^\circ=5/12. Opposite is shorter than adjacent, so the angle is less than 45∘45^\circ.

Now ask for the other acute angle of that same triangle. Opposite and adjacent swap, so the fraction reverses.

Worked example 2: the other acute angle

Question. Find the acute angle ϕ\phi opposite the 12 cm side of the same 5 cm by 12 cm right triangle.

Plan. Use tangent again, relabelling the two shorter sides relative to ϕ\phi.

tan⁡ϕ=125ϕ=tan⁡−1(125)=67.380135…∘\begin{aligned} \tan\phi&=\frac{12}{5}\\ \phi&=\tan^{-1}\left(\frac{12}{5}\right)=67.380135\ldots^\circ \end{aligned}

Answer. 67.4∘67.4^\circ to one decimal place.

Check. The unrounded angles sum to 90∘90^\circ. This also checks that the two labels refer to different acute angles.

Sine and cosine inverse inputs for an acute right-triangle angle must lie strictly between 0 and 1. An input such as 13/513/5 signals a reversed fraction or a wrongly identified hypotenuse. Tangent can take a positive input greater than 1.

Watch out

Evaluating tan⁡(5/12)\tan(5/12) gives a ratio for a tiny angle; it does not find the angle whose tangent is 5/125/12. Use tan⁡−1(5/12)\tan^{-1}(5/12).

Choose the function from the known sides.

Angles in degrees

The same inverse calculation can display a different number if the calculator is in another angle mode. This course uses degrees. Check DEG before calculating, and normally give a non-exact angle to one decimal place unless a question requests different accuracy.

For example, an inverse sine display of about 0.64 instead of about 36.9 for an input of 0.6 indicates radian mode. Change the mode and recalculate; do not add a degree sign to the wrong display.

Worked example 3: opposite and hypotenuse

Question. A right triangle has opposite side 6 cm and hypotenuse 10 cm relative to angle β\beta. Find β\beta to one decimal place.

Plan. Use inverse sine in degree mode and round the final angle.

sin⁡β=610=0.6β=sin⁡−1(0.6)=36.869897…∘≈36.9∘\begin{aligned} \sin\beta&=\frac{6}{10}=0.6\\ \beta&=\sin^{-1}(0.6)=36.869897\ldots^\circ\\ &\approx36.9^\circ \end{aligned}

Answer. 36.9∘36.9^\circ.

Check. sin⁡36.869897…∘=0.6\sin36.869897\ldots^\circ=0.6, whereas sin⁡36.9∘\sin36.9^\circ is about 0.60042, close after rounding. The angle is acute.

The next triangle gives an adjacent side instead of an opposite side. The inverse function changes, but the degree and precision decisions stay the same.

Continue this lesson.

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