Trigonometry in the Right Triangle · Grade 10
How do you find a side length when the angle is known?
Pick the ratio linking the side you want to the side you have: sine if a hypotenuse and an opposite side are involved, cosine for a hypotenuse and an adjacent side, tangent for two legs. Write the ratio, substitute what is known, and isolate the unknown. The angle stays inside the function until the very end.
Learning objectives
- Choose which of the three ratios connects the given side to the wanted one
- Solve for a side that sits in the numerator and for one that sits in the denominator
- Complete all the sides and angles of a right triangle from two given values
Choosing the right ratio
Before any calculation, ask two questions: which side is known and which is wanted. Together the two answers pick the ratio, and there is nothing to guess.
If the two sides are a leg and the hypotenuse, it is sine or cosine depending on whether the leg is opposite the angle or beside it. If both are legs, it is tangent.
It is worth writing the three words next to the diagram before calculating: opposite, adjacent, hypotenuse. Most errors here are not arithmetic errors but a wrong choice of ratio, and they disappear once the diagram is labelled. A question that arrives without a diagram is worth a minute of drawing before you touch the calculator.
Isolating the unknown
Once you have written the ratio and substituted, what remains is a simple equation. If the unknown is on top, multiply both sides; if it is underneath, multiply and then divide.
The angle itself does not change during the rearranging. The sine of thirty is a number, and until you substitute its value you can treat it as a number like any other.
Checking the answer is sensible
The hypotenuse must come out longer than either leg. If you got a shorter hypotenuse, you chose the inverse ratio or substituted in the wrong place.
A second check: the larger the angle, the longer the side facing it. An answer contradicting that points to a swap between opposite and adjacent.
Worked examples
In a right triangle the hypotenuse is 10 and the angle is 30 degrees. What is the side opposite the angle?
- A hypotenuse and an opposite side are involved, so sine
- Sine 30 equals the side over 10
- Sine 30 is 0.5
- Multiply: 0.5 times 10
Answer: The opposite side is 5
In the same triangle, what is the side adjacent to the angle?
- A hypotenuse and an adjacent side are involved, so cosine
- Cosine 30 is about 0.866
- Multiply by 10
Answer: The adjacent side is about 8.66
The side adjacent to a 40 degree angle is 6. What is the opposite side?
- Two legs, so tangent
- Tangent 40 equals the opposite side over 6
- Tangent 40 is about 0.839
- Multiply by 6
Answer: The opposite side is about 5.03
Common mistakes
- Dividing when you should multiply
- If the unknown is on top, multiply both sides by the denominator. Dividing instead shrinks the answer by the square of the ratio, which is usually obvious from the size alone.
- The calculator is set to radians
- At this level angles are in degrees. A calculator in radian mode returns numbers that look plausible and are not — sine 30 comes out as minus 0.988 instead of 0.5.
- Getting a hypotenuse shorter than a leg and carrying on
- The hypotenuse is always the longest side. A result contradicting that means the chosen ratio or the substitution is reversed, and there is no point continuing to part two.
What to remember
- The known and wanted sides choose the ratio.
- Leg and hypotenuse — sine or cosine. Two legs — tangent.
- The hypotenuse must come out longest.
- Check the calculator is in degrees.
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 an angle when two sides are 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?