Trigonometry in the Right Triangle · Grade 10
What are the special angles and what is the fundamental trigonometric identity?
The special angles are thirty, forty-five and sixty degrees, whose ratios come from two simple triangles and are worth knowing by heart. The fundamental identity says that sine squared plus cosine squared equals one — it follows directly from Pythagoras' theorem and lets you find one ratio from the other.
Learning objectives
- Derive the ratios of thirty, forty five and sixty degrees from half an equilateral triangle and from an isosceles right triangle
- Use the identity that sine squared plus cosine squared equals one
- Find the cosine of an angle when its sine is given, without a calculator
Where the values come from
Forty-five degrees comes from a right isosceles triangle: the two legs are equal and the hypotenuse is larger by a factor of root two. So sine and cosine are equal there, and the tangent is exactly one.
Thirty and sixty come from an equilateral triangle cut in half. The cut creates a right triangle in which the short leg is half the hypotenuse — and hence the sine of thirty is exactly a half.
What is worth remembering
The sine of thirty is a half, and the cosine of sixty is a half. The sine of forty-five equals the cosine of forty-five. The tangent of forty-five is one.
The pattern is worth more than the numbers: the sine of an angle equals the cosine of its complement to ninety. That is why thirty and sixty always appear together.
It is worth sketching the two triangles in the margin of the rough paper at the start of any exam. They take under a minute, and they return every value for thirty, forty-five and sixty without relying on memory in a tense moment.
The fundamental identity
Write the sine and the cosine of the same angle as ratios and square them. The sum is the sum of the squares of the legs over the hypotenuse squared — and by Pythagoras the numerator equals the denominator.
So sine squared plus cosine squared equals one, for every angle. This is what lets you find the cosine when the sine is given, without knowing the angle at all.
Worked examples
Find the side opposite a 30 degree angle when the hypotenuse is 12
- The sine of 30 is a half
- Multiply: a half times 12
Answer: The side is 6
The sine of an acute angle is 0.6. What is the cosine?
- By the identity: cosine squared equals one minus 0.36
- That is 0.64
- Take the root, positive because the angle is acute
Answer: The cosine is 0.8
What is the tangent of 45 degrees, and why?
- In a right isosceles triangle the two legs are equal
- The tangent is opposite over adjacent
Answer: The tangent of 45 is 1
Common mistakes
- Writing sine of x squared instead of sine squared of x
- Sine squared means squaring the result, not the angle. The sine of 30, squared, is 0.25, whereas the sine of 900 is something else entirely.
- Forgetting the root in the identity can be negative
- For an acute angle the cosine is positive, so you take the positive root. Later, for obtuse angles, the sign is set by the quadrant — and forgetting this becomes expensive there.
- Trying to memorise a whole table
- The three special angles come from just two triangles. Anyone who remembers the triangles can reconstruct any value in thirty seconds, which is safer than a memorised list.
What to remember
- 30, 45 and 60 come from two simple triangles.
- Sine of an angle equals cosine of its complement.
- Sine squared plus cosine squared equals one — from Pythagoras.
- The identity gives one ratio from the other.