Exact Trig Values

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Table of contents
Exact Trig Values (CIE IGCSE Maths: Extended)
Revision Note
Understanding Trigonometric Exact Values
Special Right-Angled Triangles
Images
Pythagoras Theorem and Trigonometry
Practical Application of Trigonometric Values
Sketching Triangles and Trigonometric Ratios

Exact Trig Values (CIE IGCSE Maths: Extended)

Revision Note

Author: Expertise in Maths

What are exact values in trigonometry?

Exact values in trigonometry refer to specific values of sine, cosine, and tangent for certain angles that can be expressed precisely using fractions and surds.
You should be able to recognize and derive these values geometrically.
It is essential to know the exact values for angles of 0°, 30°, 45°, 60°, 90°, 180°, and their multiples.

Summary of Exact Trig Values:

For certain angles, sin θ, cos θ, and tan θ can be expressed precisely using fractions and surds.
Familiarity with these values and the ability to derive them geometrically is crucial.

Examples:

For example, the exact value of sin 30° is 1/2, cos 60° is 1/2, and tan 45° is 1.
Understanding these values aids in solving trigonometric equations and geometric problems.

Understanding Trigonometric Exact Values

Trigonometric Values Using Fractions and Surds
Deriving Values Using Geometry
Exact Values of sin, cos, and tan
The Concept of Exact Trigonometric Values

Trigonometric Values Using Fractions and Surds

Trigonometry often involves working with fractions and surds, which are expressions involving square roots.

Deriving Values Using Geometry

Understanding trigonometric values requires a grasp of geometry concepts to derive these values accurately.

Exact Values of sin, cos, and tan

It is crucial to memorize the exact values of sine, cosine, and tangent for specific angles.These angles include 0°, 30°, 45°, 60°, 90°, 180°, and their multiples.

Angle (θ)	Sin(θ)	Cos(θ)	Tan(θ)
0°	0	1	0
30°	1/2√3	1/2	√3/3
45°	1/2√2	1/2√2	1
60°	√3/2	1/2	√3
90°	1	0	Undefined
180°	0	-1	0

The Concept of Exact Trigonometric Values

Trigonometric values for specific angles are derived based on fundamental principles of trigonometry.

Special Right-Angled Triangles

Two special right-angled triangles contain all necessary exact values
One triangle: Hypotenuse of 2 units, shorter side of 1 unit

Using Pythagoras’ theorem, the third side can be calculated
Angles: 90°, 60°, and 30°
SOHCAHTOA: Sin 60° = Sin 30° = Cos 60° = Cos 30° = Tan 60° = Tan 30°

Another triangle: Isosceles with equal sides of 1 unit each

Pythagoras’ theorem yields the hypotenuse length
Angles: Two 45° angles
SOHCAHTOA: Sin = Cos = Tan = 1

Explaining the Special Right-Angled Triangles

Special right-angled triangles offer a convenient way to find exact trigonometric values without extensive calculations. Let's delve into each triangle for a clearer understanding:

Triangle with Hypotenuse of 2 units and Shorter Side of 1 unit

In this triangle, the hypotenuse is 2 units, and one side is 1 unit. By applying Pythagoras’ theorem, we can determine the length of the third side. The angles in this triangle are fixed at 90°, 60°, and 30°. Using the mnemonic SOHCAHTOA, we find that Sin 60° = Sin 30° = Cos 60° = Cos 30° = Tan 60° = Tan 30°.

Isosceles Triangle with Equal Sides of 1 unit

This isosceles triangle has two equal sides of 1 unit each. According to Pythagoras’ theorem, the hypotenuse length can be calculated. The two equal angles in this triangle are both 45°. When we apply SOHCAHTOA, we discover that Sin = Cos = Tan = 1.

Images

Pythagoras Theorem and Trigonometry

When we apply Pythagoras' theorem to a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Exact Trig Values - Year 11 PDF Download (21)

In a right-angled triangle, if the two non-right angles are equal, they are each 45 degrees.

Exact Trig Values - Year 11 PDF Download (22)

SOHCAHTOA is a mnemonic device to remember the trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Exact Trig Values - Year 11 PDF Download (23)

Exact Trig Values - Year 11 PDF Download (24)

Practical Application of Trigonometric Values

It's crucial to be comfortable with exact trigonometric values for specific angles, especially during exams.
Sketching triangles can help in visualizing these values and angles effectively.

Exact Trig Values - Year 11 PDF Download (25)

Exact Trig Values - Year 11 PDF Download (26)

Exact Trig Values - Year 11 PDF Download (27)

For instance, in an equilateral triangle with side length 2 units, we can derive exact values for sine, cosine, and tangent of 60 and 30 degrees.

Sketching Triangles and Trigonometric Ratios

Begin by sketching a triangle and creating two right-angled triangles using the line of symmetry through the middle.
Utilize Pythagoras' theorem to determine the vertical height of the triangle.
Apply SOHCAHTOA to calculate the trigonometric ratios for 30° and 60°.

Sine of 60° is equal to Sine of 30°.
Cosine of 60° is equal to Cosine of 30°.
Tangent of 60° is equal to Tangent of 30°.

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