Sec 60: Overview, Questions, Preparation

In mathematics, we know that a right angle triangle has three sides and three angles. Among those three angles, one angle is 90°, and the three sides are Hypotenuse, Perpendicular, and Base. The ratio of the hypotenuse and the base are known as the secant of the angle. Suppose we have angle θ in the right-angled triangle, then the secant for θ will be 

sec θ = Hypotenuse/Base

sec θ is the reciprocal of cos θ hence it can also be written as 1/cos θ.

For finding the value of sec 60°, we are considering an equilateral triangle ABC. Therefore every angle will be 60°( ∠A=∠ B=∠ C=60°).

Now, from A, we will draw a perpendicular line AD at BC

Therefore △ ABD ≅ △ACD and BD = DC,

If BD = DC then ∠ BAD = ∠ CAD = 30°,

Now, let us assume the lengths of each side to be 2a,

Therefore AB = BC = CA = 2a and BD = DC = a,

Now we'll find the value of cos 60°,

cos θ = 1/sec θ = Adjacent Side / Hypotenuse 

Now θ = 60°

Cos 60°= BD/AB = a/2a

Hence cos 60° = 1/2,

Now, we know that the reciprocal of cosine is secant

Therefore sec 60° = 1/cos 60° = 2.

Secant is the trigonometry angle; apart from that, there are also other trigonometric operators like sin, cos, cot, tan, cosec. The trigonometric values from 0 to 90 are:

Secant is the topic of the chapter Introduction to Trigonometry of class X mathematics. The chapter is involved in various fields of mathematics. Every year around 5 to 6 questions ranging from easy to moderate level are asked from the chapter. The weightage of these questions in class X exams is 10 to 12 marks. Introduction to trigonometry also includes topics like:

• Trigonometric Ratios of complementary and specific angles
• Trigonometric Identities
• Measurement of height and distances using trigonometric applications

1. Suppose in a right-angled triangle the XYZ, the base's length is 21cm and Hypotenuse's length is 29 cm. Then find the value of cos2 + sin2?

From applying the Pythagoras theorem, we will get the perpendicular's length, i.e., 20cm.

Now cosθ = 21/29 and sinθ = 20/29,

Therefore cos2 + sin2 = (21/29)2 + (20/29)2 = 1.

2. Write cot 85° + cos 75° in the manner of 0° to 45°. 

We know that cot θ can be written as tan(90- θ) and cos θ as sin(90- θ).

Therefore, cot 85° + cos 75° = tan(90-85) + sin(90-75)

= tan 5° + sin 15°.

3. Prove that sec θ (sec θ + tan θ) (1 -sin θ) = 1.

Taking LHS, 

sec θ(1 -sin θ)(sec θ + tan θ) 

= (1/cos θ)*(1/cos θ + sinθ/cos θ)*(1-sin θ)

= (1+sinθ)*(1-sin θ)/cos2 θ 

= (1-sin2 θ)/cos2 θ 

= 1.

Q1. What is the reciprocal of tan θ?

Q2. Mention the undefined trigonometric angles from the first quadrant?

Q3. cos A + cos B equals to?

Q4. What is the value of sec 15?

Q5. Write sec θ in the manner of cosec?