Solve right triangles
To solve a right triangle means to calculate the sides and angles from the information given.
If a triangle has a right-angle then in order to determine all the sides and angles it is sufficient to know either two sides or one side and one angle other than the right-angle.
For the solution, the mathematical tools needed may include Pythagoras's theorem, the trigonometric functions (sine, cosine and tangent) and the inverse trigonometric functions.
Worked Examples
Question 1
A right triangle is given in which the hypotenuse has length 1553 and another side length is 495 units. Find the third side and the two non right-angles.
By Pythagoras, the missing side is $\sqrt{1553^2-495^2}=1472$√15532−4952=1472.
According to the given information, $\cos A=\frac{495}{1553}\approx0.31874$cosA=4951553≈0.31874. To find the angle whose cosine is $0.31874$0.31874 we use the inverse cosine function:
$A=\cos^{-1}0.31874\approx71.4^\circ$A=cos−10.31874≈71.4°
We could use the inverse sine function to find angle $B$B but it is simpler to use the fact that $A+B=90^\circ$A+B=90°. Thus,
$B=90^\circ-71.4^\circ=18.6^\circ$B=90°−71.4°=18.6°.
Question 2
In the given triangle, find angle $A$A and the sides $a$a and $c$c.
Since angle $A$A is the complement of the $18.6^\circ$18.6° angle, we see that $A=90^\circ-18.6^\circ=71.4^\circ$A=90°−18.6°=71.4°.
Focussing on the given angle, we can use the fact that $\tan18.6^\circ=\frac{495}{a}$tan18.6°=495a. Thus,
$a=\frac{495}{\tan18.6^\circ}\approx1471$a=495tan18.6°≈1471.
We could now use Pythagoras's theorem to find $c$c, or we could use the sine function: $\sin18.6^\circ=\frac{495}{c}$sin18.6°=495c. So,
$c=\frac{495}{\sin18.6^\circ}\approx1552$c=495sin18.6°≈1552.
The differences between these results and those in the first example can be explained as round-off errors.
More Worked Examples
QUESTION 3
Consider the diagram.
Find $\sin A$sinA.
Form an expression for $\cos A$cosA.
Form an expression for $\tan A$tanA.
Form an expression for $\csc\left(A\right)$csc(A).
Form an expression for $\sec\left(A\right)$sec(A).
Form an expression for $\cot\left(A\right)$cot(A).
QUESTION 4
Consider $\triangle ABC$△ABC in which $\angle ABC=45^\circ$∠ABC=45°.
Solve for the exact length of the side $y$y. Leave your answer in rationalised form.
Solve for the exact length of the side measuring $x$x. Leave your solution in rationalised form.
QUESTION 5
Solve the triangle $ABC$ABC with $C=90^\circ$C=90°, $a=12.9$a=12.9 kilometres and $b=17.2$b=17.2 kilometres.
Solve for $c$c.
Solve for $A$A correct to two decimal places.
Solve for $B$B correct to two decimal places.