Exact Trigonometric Values
This topic covers the exact trigonometric values for key angles (0°, 30°, 45°, 60°, 90°), which you must know or be able to derive instantly. These values are fundamental building blocks for solving non-calculator problems in geometry, mechanics, and other scientific contexts.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The values for sin, cos, and tan at 0° and 90° represent the axes on the unit circle. Remember that tan(90°) is undefined because it involves division by cos(90°), which is zero.
- Values for 30° and 60° are complementary, meaning sin(30°) = cos(60°) and cos(30°) = sin(60°).
- For 45°, the sine and cosine values are equal: sin(45°) = cos(45°) = 1/√(2).
- It is essential to either memorise the table of values or be able to derive them in seconds by sketching two special triangles.
- The first derivation triangle is an isosceles right-angled triangle with shorter sides of length 1, giving a hypotenuse of √(2) for finding 45° values.
- The second is half of an equilateral triangle of side length 2, creating a right-angled triangle with sides 1, √(3), and hypotenuse 2, for finding 30° and 60° values.
Formulae
sin(30) = 1/2, cos(30) = √(3)/2, tan(30) = 1/√(3) Standard values for calculations involving 30°.
sin(45) = 1/√(2), cos(45) = 1/√(2), tan(45) = 1 Standard values for calculations involving 45°.
sin(60) = √(3)/2, cos(60) = 1/2, tan(60) = √(3) Standard values for calculations involving 60°.
tan(x) = sin(x) / cos(x) To find the value of tan(x) if you know sin(x) and cos(x), or to understand why tan(90) is undefined.
Definitions
- SOH CAH TOA
- A mnemonic for the trigonometric ratios in a right-angled triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
- Undefined
- A value that cannot be determined, such as the result of division by zero. For this topic, tan(90°) = sin(90°)/cos(90°) = 1/0, which is undefined.
Worked example
Without using a calculator, determine the exact value of: (tan(45) + sin(30)) / cos(60)2
- 1
First, substitute the known exact values into the expression.
tan(45) = 1, sin(30) = 1/2, and cos(60) = 1/2 - 2
The expression becomes:
(1 + 1/2) / (1/2)2.
- 3
Simplify the numerator:
1 + 1/2 = 3/2 - 4
Calculate the denominator:
(1/2)2 = 1/4 - 5
The expression is now a division of two fractions:
(3/2) / (1/4).
- 6
To divide by a fraction, multiply by its reciprocal:
(3/2) × (4/1).
- 7
Perform the multiplication:
(3 × 4) / (2 × 1) = 12 / 2 = 6.
Answer: 6
Common mistakes
- ×Errors in arithmetic after substituting the correct values are common. Be extra careful when squaring fractions or dealing with expressions involving square roots.
- ×Mixing up the values for sin(30) and sin(60), or for cos(30) and cos(60). Remember sin increases from 0 to 1 as the angle goes from 0 to 90 degrees.
- ×Forgetting that tan(90) is undefined, and treating it as zero or one in a calculation.
No-calculator tips
- ✓If you forget a value, quickly sketch the 1-1-√(2) and 1-√(3)-2 triangles. It takes 5 seconds and guarantees accuracy.
- ✓Remember the simple pattern for sin(x) values: √(0)/2, √(1)/2, √(2)/2, √(3)/2, √(4)/2 for x = 0, 30, 45, 60, 90. Cosine is the same pattern in reverse.
- ✓When simplifying fractions like 1/√(2), you can often leave the root in the denominator if the final calculation becomes simpler, but be prepared to rationalise it to `√(2)/2` if needed.