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Find the value of cos(405)°

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  • Last Updated : 03 Sep, 2021
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Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. The six main trigonometric functions are sine, cosine, tangent, cotangent, secant, or cosecant.

Angles defined by the ratios of trigonometric functions are known as trigonometry angles. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.

Trigonometric Identities

Equalities involving trigonometric functions which are true for every value of the occurring variables for which both sides of the equality are defined are known as trigonometric identities. The various types of trigonometric identities are as follows,

Reciprocal Identities

 \newline \sin \theta= \frac{1}{\cosec \theta} \newline \cos \theta= \frac{1}{\sec \theta} \newline \tan \theta= \frac{1}{\cot \theta} \newline \cot \theta= \frac{1}{\tan \theta} \newline \sec \theta= \frac{1}{\cos \theta} \newline \cosec \theta= \frac{1}{\sin \theta}

Pythagorean Trigonometric Identities

\sin^2θ + \cos^2θ = 1 \newline \sec^2θ-\tan^2θ = 1 \newline \cosec^2θ-\cot^2θ = 1

Complementary and Supplementary Trigonometric Identities

Complementary angles are a pair of angles whose sum is equal to 90°. (90° -\theta  ) is the complement of an angle \theta. Following are the trigonometric ratios of complementary angles,

\sin(\frac{\pi}{2}-\theta)=\cos\theta \newline \cos(\frac{\pi}{2}-\theta)=\sin\theta \newline \tan(\frac{\pi}{2}-\theta)=\cot\theta \newline \cot(\frac{\pi}{2}-\theta)=\tan\theta \newline \sec(\frac{\pi}{2}-\theta)=\cosec\theta \newline \cosec(\frac{\pi}{2}-\theta)=\sec\theta

Supplementary angles are a pair of angles whose sum is equal to 180°. 180° -\theta is the supplement of an angle \theta. Following are the trigonometric ratios of supplementary angles,

\newline \sin(\pi-\theta)=\sin\theta \newline \cos(\pi-\theta)=-\cos\theta \newline \tan(\pi-\theta)=-\tan\theta \newline \cot(\pi-\theta)=-\cot\theta \newline \sec(\pi-\theta)=-\sec\theta \newline \cosec(\pi-\theta)=\cosec\theta

Find the value of cos(405)°

Solution:

= cos 405° = cos (360° + 45°)

= cos(2π+45°)

It is known, cos(2π+θ) = cos(θ)

So, cos(2π+45°)=cos(45°)

= cos(360°+45°)=cos(45°) =\frac{1}{√2}

Thus, the value of cos(405°) =\frac{1}{√2}

Similar Problems

Question 1: Find the value of Sin135°.

Solution:

= sin135° = sin(180° – 45°)

sin(180°−θ) = sinθ

\sin (135\degree)=\frac{1}{\sqrt{2}}

Question 2: What is the exact value of tan210°.

Solution:

= tan210° = tan(180°+30°)

We know that, tan(180°+θ)=tanθ

\tan(210\degree) =\frac{1}{\sqrt{3}}

Question 3: Find the exact value of cos225°.

Solution:

= cos225° = cos(180° + 45°)

We know that, cos(180°+θ)=−cosθ 

-\cos(225\degree) =-\frac{1}{\sqrt{2}}

Question 4: Find the value of sin330°.

Solution:

= sin330° = sin(360° – 30°)

We know that, \sin(2\pi-\theta)=-\sin\theta

-\sin330\degree=-\frac{1}{2}


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