Cos 45 Degrees Explained with Value and Derivation

The concept of cos 45 degrees is a core topic in trigonometry, often tested in maths exams and used in STEM fields. Understanding and remembering its value helps students solve a wide range of geometry and algebra problems efficiently.

What Is Cos 45 Degrees?

Cos 45 degrees is the cosine of a 45° angle. It is a trigonometric ratio that relates the length of the adjacent side to the hypotenuse in a right-angled triangle. You'll encounter this value in geometry, the unit circle, and various real-life applications like physics and engineering. In trigonometry, cos 45° is considered a "standard angle" because its value is easy to learn, use, and apply to different problems.

Key Formula for Cos 45 Degrees

Here’s the standard formula: \( \cos 45^\circ = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \)

Value of Cos 45 Degrees (Exact, Fraction, Decimal, Radian)

Angle	Fraction (Surd)	Decimal	Radian Equivalent
45°	\( \frac{\sqrt{2}}{2} \)	0.7071	\( \frac{\pi}{4} \)

How to Derive the Value of Cos 45 Degrees

Let's derive cos 45 degrees using an isosceles right triangle and the unit circle for a clear understanding.

1. Draw a right triangle where both non-right angles are 45° (isosceles right triangle).
2. Let the two shorter sides be 1 unit each.
   By Pythagoras’ theorem, hypotenuse = \( \sqrt{1^2 + 1^2} = \sqrt{2} \)
3. Cos 45° = Adjacent/Hypotenuse = \( 1/\sqrt{2} \)
4. Rationalize: \( 1/\sqrt{2} = \sqrt{2}/2 \) (exact value)

On the unit circle, the x-coordinate at 45° is also \( \sqrt{2}/2 \).

Cos 45 Degrees in Trigonometric Tables

The value of cos 45 degrees is a key entry in any trigonometric table. It is regularly used in competitive and board exams for solving MCQs, geometry, and trigonometric identities. Being able to recall cos 45° quickly allows you to solve questions faster and score better.

Angle	Cosine Value
0°	1
30°	\( \frac{\sqrt{3}}{2} \)
45°	\( \frac{\sqrt{2}}{2} \)
60°	\( \frac{1}{2} \)
90°	0

Cos 45 Formulae and Related Identities

Here are some important identities involving cos 45 degrees:

• \( \cos^2 45^\circ + \sin^2 45^\circ = 1 \)
• \( \cos 45^\circ = \sin 45^\circ \)
• \( \cos (90^\circ - 45^\circ) = \sin 45^\circ \)
• Trigonometric identities often include \(\cos 45^\circ\) in compound angles and sum/difference formulas.

Solved Examples Using Cos 45 Degrees

Here are a few sample problems to help you master cos 45 degrees value in real scenarios.

Q1. Find the value of: \( 2\sin 60^\circ - 4\cos 45^\circ \)

1. Substitute known values: \(\sin 60^\circ = \sqrt{3}/2\), \(\cos 45^\circ = \sqrt{2}/2\)

2. Calculate: \(2 \times (\sqrt{3}/2) - 4 \times (\sqrt{2}/2)\)

3. Simplify: \(\sqrt{3} - 2\sqrt{2}\)

4. Final Answer: \(\sqrt{3} - 2\sqrt{2}\)

Q2. What is \( \cos 45^\circ + \cos 90^\circ \)?

1. Substitute: \(\cos 45^\circ = \sqrt{2}/2\), \(\cos 90^\circ = 0\)

2. Add: \(\sqrt{2}/2 + 0 = \sqrt{2}/2\)

3. Final Answer: \(\sqrt{2}/2\)

Q3. Evaluate: \(3\cos 45^\circ + 2\sin 60^\circ\)

1. Known values: \(\cos 45^\circ = \sqrt{2}/2\); \(\sin 60^\circ = \sqrt{3}/2\)

2. Multiply: \(3 \times (\sqrt{2}/2) + 2 \times (\sqrt{3}/2)\)

3. Simplify: \((3\sqrt{2} + 2\sqrt{3})/2\)

4. Final Answer: \((3\sqrt{2} + 2\sqrt{3})/2\)

Speed Trick to Remember Cos 45 Degrees

A simple trick for remembering the cosine values of standard angles (0°, 30°, 45°, 60°, and 90°) is to use the square root pattern:

• \(\cos \theta = \sqrt{n}/2\) where n = 4, 3, 2, 1, 0 for 0°, 30°, 45°, 60°, 90°, respectively.

Thus, for cos 45°, n = 2: \(\cos 45^\circ = \sqrt{2}/2\).

Relation to Other Trigonometric Concepts

Learning cos 45 degrees helps you connect with important topics like sin 45 degrees (which equals cos 45°), the trignometric table, and trigonometric ratios for other standard angles.

Wrapping It All Up

We explored cos 45 degrees—its definition, derivation, conversion to radian, decimal and fraction, usage in tables and examples, along with memory tricks. Keep practicing and refer to Vedantu's resources for doubt clearance and more interactive learning on trigonometry.

Try These Yourself

• Calculate cos 45 degrees as a decimal without a calculator.
• Prove cos 45 = sin 45 using a right triangle.
• Find the area of an isosceles right triangle with legs of length 5 using cos 45°.
• Solve: If cos θ = cos 45°, what is θ between 0° and 360°?

Internal Links for Further Learning

• Sin 45 Degrees: Understand why sin 45° = cos 45°.
• Trigonometry Table: Look up all major angles and their sine, cosine, and tangent values.
• Unit Circle: Visual guide to trig ratios and their coordinates.
• Trigonometric Functions: Dive deeper into trigonometric graphs, domains, and inverses.
• Degrees to Radians: Convert 45° to radians and vice versa for advanced applications.

Continue your maths journey with Vedantu to master cos 45 degrees and all other important trigonometric values for exams and real life!