Worksheet Solution: Introduction to Trigonometry

True and False

Q1: cos A ≤ 1 for every acute angle A
Ans: True
Explanation: For any angle in a right-angled triangle, the hypotenuse is the longest side.

Since the numerator is always smaller than or equal to the denominator, cosA ≤ 1

So this is always true for acute angles.

Q2: tanA = sinA/cosA
Ans: True
Explanation: By definition, tan A = opposite/adjacent. Using sine and cosine definitions, tan A = (sin A)/(cos A), provided cos A ≠ 0.
Q3: secA = 1cosA, for an acute angle
Ans: True
Explanation: Secant is defined as the reciprocal of cosine: sec A = 1/cos A for any angle where cos A ≠ 0. This holds for acute angles as well. (Note: The correct expression is sec A = 1/cos A.)
Q4: sin60º = 2sin30º
Ans: False
Explanation: sin 60º = √3/2 and sin 30º = 1/2. So 2·sin 30º = 2·(1/2) = 1, which is not equal to √3/2. Hence the statement is false.
Q5: SinA + CosA = 1
Ans: False
Explanation: This is not true for all angles. For example, at A = 45º, sin 45º + cos 45º = √2/2 + √2/2 = √2 ≠ 1. Thus the equality does not hold generally.

Short Answer Questions

Q6: Write the values cos 0°, cos 45°, cos 60° and cos 90°. What happens to the values of cos as angle increases from 0° to 90°?
Ans: cos 0^∘ = 1
cos 45^∘ = 1/√2
cos 60^∘ = 1/2
cos 90^∘ = 0
We can see that the values of cos decrease as the angle increases from 0° to 90°.
Q7: Write the values of tan 0°, tan 30°, tan 45°, tan 60° and tan 90°. What happens to the values of tan as angle increases from 0° to 90°?
Ans: tan 0^∘ = 0
tan 30^∘ = 1/√3
tan 45^∘ = 1
tan 60^∘ = √3
tan 90^∘ = undefined (cos 90º = 0, so tan is not defined)
As the angle increases from 0° to 90°, tan increases from 0 towards infinity and becomes undefined at 90°.
Q8: If cosec A = √10 . find other five trigonometric ratios.
Ans:

Now By Pythagoras theorem

Q9: The value of (sin 30^∘ + cos 30^∘) - (sin 60^∘ + cos 60^∘) is
Ans:

sin 30º = 1/2, cos 30º = √3/2 ⇒ sin 30º + cos 30º = (1 + √3)/2.

sin 60º = √3/2, cos 60º = 1/2 ⇒ sin 60º + cos 60º = (√3 + 1)/2.

Subtracting: (1 + √3)/2 - (√3 + 1)/2 = 0.

Therefore the value is 0.

Q10: Evaluate the following: 2sin² 30^∘ - 3cos²45^∘ + tan²60^∘
Ans:

sin 30º = 1/2 ⇒ sin² 30º = 1/4 ⇒ 2 · sin² 30º = 2 · 1/4 = 1/2.

cos 45º = 1/√2 ⇒ cos² 45º = 1/2 ⇒ -3 · cos² 45º = -3 · 1/2 = -3/2.

tan 60º = √3 ⇒ tan² 60º = 3.

Sum = 1/2 - 3/2 + 3 = (1/2 - 3/2) + 3 = -1 + 3 = 2.

Therefore the value is 2.

Q11: Evaluate:
cot²30^∘  - 2cos²60^∘ - 3/4sec²45^∘ - 4sec²30^∘
Ans:

cot 30º = √3 ⇒ cot² 30º = 3.

cos 60º = 1/2 ⇒ cos² 60º = 1/4 ⇒ -2·cos² 60º = -2 · 1/4 = -1/2.

sec 45º = √2 ⇒ sec² 45º = 2 ⇒ -(3/4)·sec² 45º = -(3/4)·2 = -3/2.

sec 30º = 1/cos 30º = 1 / (√3/2) = 2/√3 ⇒ sec² 30º = 4/3 ⇒ -4·sec² 30º = -4 · 4/3 = -16/3.

Now add: 3 - 1/2 - 3/2 - 16/3 = 3 - 2 - 16/3 = 1 - 16/3 = (3/3 - 16/3) = -13/3.

Therefore the value is -13/3.

Q12: Write the values of sin 0°, sin 30°, sin 45°, sin 60° and sin 90°. What happens to the values of sin as angle increases from 0° to 90°?
Ans:

sin 90^∘ = 1, We can see that the values of sin increase as the angle increases from 0° to 90°.

Q13: If sin A = 3/5 .find cos A and tan A.
Ans:

Q14: In a right triangle ABC right angled at B if sinA = 3/5. find all the six trigonometric ratios of C.
Ans:

Q15:  If cos B = 5/13, find sin B and tan B.
Ans: We Know that,

sin²B + cos²B = 1

Now, substituting the value of cosB

We know that,