Trigonometry in the Right Triangle · Grade 10
How do you find an angle when two sides are known?
Work out the ratio between the two known sides, then apply the inverse function on the calculator — inverse sine, inverse cosine or inverse tangent, according to which sides they are. The ratio is a number, and the inverse function translates it back into the angle that produced it.
Learning objectives
- Decide which ratio to use according to the two sides that are given
- Use the inverse trigonometric keys to find an angle in degrees
- Find both acute angles of a right triangle and check that the three angles sum to one hundred and eighty degrees
The other direction
Until now the angle was known and a side was missing. Here it is the reverse: two sides are known and the angle is missing, so the calculation runs the other way.
The first step is the same — choose the ratio according to which two sides you have. The second step is the new one: instead of substituting an angle and getting a number, you substitute a number and get an angle.
What the button does
The button marked sine with an exponent of minus one is not one over sine. It is the inverse function: it takes a ratio and returns the angle that ratio belongs to.
So the order is fixed — work out the ratio as a number first, and only then apply the inverse function to it. Applying it to a single side length is meaningless.
Before pressing the button it is worth asking what answer would be reasonable. A ratio of a half belongs to an angle near thirty degrees, and a ratio near one to an angle near ninety. That estimate takes a second and catches almost any mistyped entry.
Checking the answer
An acute angle in a right triangle must fall between zero and ninety degrees. A result outside that range means an error in substitution or a calculator in radians.
Another good check: the larger ratio belongs to the larger angle. If the ratio came out near one, the angle is near ninety.
Worked examples
The side opposite the angle is 3 and the hypotenuse is 5. What is the angle?
- An opposite side and a hypotenuse, so sine
- The ratio: 3 over 5, that is 0.6
- Apply inverse sine to 0.6
Answer: The angle is about 36.9 degrees
Both legs are 4 and 4. What is the angle?
- Two legs, so tangent
- The ratio: 4 over 4, that is 1
- Apply inverse tangent to 1
Answer: The angle is 45 degrees — an isosceles right triangle
The adjacent side is 8 and the hypotenuse is 10. What is the angle?
- An adjacent side and a hypotenuse, so cosine
- The ratio: 8 over 10, that is 0.8
- Apply inverse cosine
Answer: The angle is about 36.9 degrees
Common mistakes
- Reading the exponent of minus one as one over the function
- That notation means an inverse function, not a reciprocal. One over sine is something else entirely, and using it gives a number that is not an angle at all.
- Applying the inverse function to a side instead of to the ratio
- The inverse function takes a number between minus one and one for sine and cosine. A side length such as 8 returns an error on the calculator, and that is the clue the ratio was never worked out.
- Getting an angle above 90 and carrying on
- Both acute angles in a right triangle are under ninety. A larger result means the calculator is in radians or the substitution is reversed.
What to remember
- Work out the ratio first, the angle second.
- The minus one exponent means inverse, not reciprocal.
- The answer must fall between 0 and 90 degrees.
- A larger ratio belongs to a larger angle.
More in Trigonometry in the Right Triangle
- Why does the ratio between sides depend only on the angle?
- What are sine, cosine and tangent?
- How do you find a side length when the angle is known?
- What are the special angles and what is the fundamental trigonometric identity?
- What is the difference between an angle of elevation and an angle of depression?