Content Curator
Value of Cos 60 degrees is ½. Trigonometry is a mathematical application that deals with angles, their properties and usage. It is a function that relates the angles and the sides of a right-angled triangle in solving real-life problems. It is used in finding an angle or side whose value is unknown. Many formulas and identities of trigonometry are considered when finding the value of the unknown. Trigonometric functions include the sine, cosine, tangent, cosecant, secant and cotangent.
| Table of Content |
Keyterms: Trigonometry, Trigonometric functions, Sine, Cosine, Tangent, Cosecant, Secant, Cotangent, Hypotenuse, Adjacent side
What is a Cosine Function?
[Click Here for Sample Questions]
Cosine function is defined as the relation between the adjacent side and the hypotenuse as the angle cosine is found in between the adjacent side and hypotenuse. It is given by,
Cos x = \(\frac{Adjacent}{Hypotenuse}\)
Cos Formula
Also Check: Difference between Trigonometry and Geometry
Cos 60° Value Derivation
[Click Here for Sample Questions]
Consider an equilateral triangle,
Equilateral Triangle
Here, PQ = QR = 2 units
Let S be the midpoint of the side QR.
Therefore, QS = SR = 1 unit
We know that,
Cos x = QS/PQ
Cos 60° = 1/2
Similarly, using the Pythagoras theorem,
PQ2 = PS2+QS2
→ 22 = PS2 +12
→ PS = \( {\sqrt{3}}\)
We know that,
Sin 60° = PS/PQ
Sin 60° = \( {\sqrt{\frac{3}{2}}}\)
Tan 60° = PS/QS
Tan 60° = \( {\sqrt{3}}\)
Similarly, all the trigonometric values for different angles can be found.
Also Read: Remainder Theorem
Derivation of Cos 60 by Trigonometric Method
[Click Here for Sample Questions]
Value of cos 60°, can also be found using the sine function,
Sin(90°- θ)= Cos θ
Therefore,
sin(90°- 60°) = cos 60°
→ Sin 30° = Cos 60°
From the table,
Sin 30°= 1/2
Therefore,
Cos 60°= 1/2
Also Read: Permutations and Combinations
Application And Uses Of Trigonometry
[Click Here for Sample Questions]
- Aviation, Criminology, Marine Biology, And Navigation make use of Trigonometry.
- The height of a building or a mountain is measured using trigonometry.
- In oceanography, trigonometry is used to compute the heights of waves and tides.
- It is a tool that is used to make maps and is a major component in satellite systems.
Check Out:
Things To Remember
- Radius of a unit circle is always one.
- The trigonometric function and values deal with relation to the right-angled triangle.
- The value of Cos 60 degrees is ½.
- Trigonometry is used in finding an angle or side whose value is unknown.
- According To Pythagoras Theorem: Hypotenuse2 = Base2+Altitude2
- Aviation, Criminology, Marine Biology, And Navigation make use of Trigonometry.
Also Read: Derivative of Inverse Trignometric Functions
Sample Question
Ques. What are the identities of cos functions? (2 marks)
Ans. cos(-x) = cos x
cos(x+2nπ) = cos x
sin2 x + cos2 x = 1
cos(x+y) = cosx.cosy–sinx.siny
cos(x–y) = cosx.cosy+sinx.siny
Ques. If θ= 60°, find the value of tan θ using cos and sin. (2 marks)
Ans. Sin 60° = \( {\sqrt{\frac{3}{2}}}\)
Cos 60°= 1/2
Tan 60°= Sin 60°/Cos 60°
Tan 60°= \( {\sqrt{3}}\)
Ques. What is the relation between the quadrant and the trigonometric sign? (2 marks)
Ans. Quadrant I: All functions are positive.
Quadrant II: Only sine functions are positive
Quadrant III: Only tangent functions are positive
Quadrant IV: Only cosine functions are positive
Ques. What is the sign of Cos 400°? (2 marks)
Ans. Cos 400° = Cos (360°+4°)
We infer that Cos 400 belongs to the first quadrant, therefore the sign of Cos 400 is positive.
Ques. Find the value of Cos 30° + Sin 60° (2 marks)
Ans. Let Cos 30° + Sin 60° be equation (1)
Sin60° = Sin (90° - 30°)
Sin60° = Cos30°
Now, equation (1) becomes,
Cos 30° + Cos 30° = \( {\sqrt{\frac{3}{2}}}\) + \( {\sqrt{\frac{3}{2}}}\) = 2 \( {\sqrt{\frac{3}{2}}}\)
Ques. If a = xsinθ and p = xcosθ, what is the value of a2+p2? (2 marks)
Ans. a2+p2
=(xsinθ)2+(xcosθ)2
=x2 (sin2θ+cos2θ)
=x2
Ques. What is the value of cos 60° in rad? (2 marks)
Ans. Cos 60° = 1/2
Cos 60° = 60 x π/180
=1/3 π
Ques. What is the decimal value of cos 60°? (1 mark)
Ans. The value of Cos 60° is 1/2, therefore, the decimal value of Cos 60° is 0.5.
Ques. If A, B and C are the interior angles of a ΔABC, show that sin (A+B²) = cos(c²). (CBSE 2012) (2 marks)
Ans. In ?ABC, ∠A + ∠B + ∠C = 180° …(Angle sum property of Δ)
∠A + ∠B = 180° – ∠C
\(\frac{\angle A + \angle B}{2} = \frac{180^o - \angle C}{2} = 90^o - \frac{\angle C}{2}\) ...(i)
...[divided by 2]
L.H.S = sin\((\frac{\angle A + \angle B}{2})\)
= sin \((90^o - \frac{\angle C}{2})\) ...[From (i)]
= cos \(\frac{\angle C}{2}\) = R.H.S
\(\Rightarrow\) L.H.S = R.H.S
Ques. If x = a cos θ – b sin θ and y = a sin θ + b cos θ, then prove that a2 + b2 = x2 + y2. (CBSE 2015) (2 marks)
Ans. R.H.S. = x² + y²
= (a cos θ – b sin θ)² + (a sin θ + b cos θ)²
= a² cos² θ + b² sin² θ – 2ab cos θ sin θ + a² sin² θ + b² cos² θ + 2ab sin θ cos θ
= a² (cos² θ + sin²θ) + b² (sin² θ + cos² θ)
= a² + b² = L.H.S. …[\(\because \) cos² θ + sin² θ = 1]
Ques. If cos x = cos 40° . sin 50° + sin 40°. cos 50°, then find the value of x. (CBSE 2014) (2 marks)
Ans. cos x = cos 40° sin 50° + sin 40° cos 50°
cos x = cos 40° sin(90° – 40°) + sin 40°.cos(90° – 40°)
cos x = cos² 40° + sin² 40°
cos x = 1 …[\(\because \)cos² A + sin² A = 1
cos x = cos 0° ⇒ x = 0°
Ques. Prove the identity: (sec A – cos A). (cot A + tan A) = tan A . sec A. (CBSE 2014) (2 marks)
Ans. L.H.S.= (sec A – cos A) (cot A + tan A)
= \((\frac{1}{cos A} - cos A)(\frac{cosA}{sinA} + \frac {sinA}{cosA})\)
= \((\frac{1 - cos^2 A}{cos A})(\frac{cos^2 A + sin^2 A}{sinAcosA})\)
= \(\frac{sin^2}{cos A} \times \frac{1}{sinAcosA} = \frac{sinA}{cosA} \times \frac{1}{cosA}\)
…[\(\because cos^2A + sin^2 A = 1\)]
= tan A . sec A = R.H.S.
Ques. If θ = 30°, verify the following: (CBSE 2014) (4 marks)
(i) cos 3θ = 4 cos³ θ – 3 cos θ
(ii) sin 3θ = 3 sin θ – 4 sin³θ
Ans.
Check-Out:
Comments