Most Important Questions Class 10 Maths Ch 8 Trigonometry with Solutions

Mathematics Trigonometry Important Questions Class 10 contains various questions that are useful for students even in their competitive examinations like JEE & NEET. Many students consider trigonometry as one of the electrifying and tough chapters in mathematics. To begin with, the subject may look like a collection of unfamiliar angles, triangles, and ratios. But once one understands the basic principles of trigonometry, then it is no longer merely an area in examinations but a practical part of the mathematics utilised in day-to-day activities. 

We aim to clarify the significance of these questions for you. Here at Educart, we will delve deeper into the important questions related to trigonometry in class 10. We will also delve into the fundamentals of trigonometry, explore its application for exam success, and explore its benefits beyond academics.

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Chapter 8  Introduction to Trigonometry: Important Questions

1. If cos y = 0, then what is the value of ½ cos y/2?

1. 0
2. ½
3. 1/√2
4. 1/2√2

Soln.

‍d) 1/2√2

Explanation: 

cos y = 0 

let y= π/2 + kπ where k is an integer

Substitute y= π/2 + kπ in ½ cos y/2

= ½ cos (π/2 + kπ)/2

= ½ cos (π/4) + (kπ/2)

put k=0

= ½ cos (π/4) + ((0)π/2)

= ½ cos (π/4) 

= ½ . 1/√2

=1/2√2

2. Show that sinθ  = cos (90-θ ) is true using the definition of trigonometric ratios.

Soln.

∠PQR = 90°

∠QRP = 90°- θ

the sum of angles in a triangle is 180 and one angle is 90°

sin θ = PQ/PR

sin θ = ⅘

cos(90°- θ) = QP/PR

cos(90°- θ) = ⅘

(In the right triangle, the side opposite to θ is the same as the side adjacent to 90°−θ. Therefore, sin(θ)=cos(90°−θ).

Hence, sin(θ)=cos(90°−θ) is true

3. In the triangles shown below, ∠Q = ∠T.

Write an expression each for cos Q and sin T.

Soln.

‍In △PQR

The side adjacent to ∠Q is r, and the hypotenuse is p.

cosQ=r/p

In △TUS:​

sinT=t/s

4. A unit circle is shown below with centre O. A tangent AB is drawn to the circle at point  M such that ∠MOB = β.

(Note: The figure is not to scale.)

If OA ⊥ OB, write the expressions that represent the lengths of

i) OB

ii) OA 

iii) AB

Soln. 

i) OB

The unit circle has a radius of 1 unit, centered at O.

AB is a tangent at point M.

∠MOB=β

OA is perpendicular to OB

Length of OB:

In the right triangle △OMB:

• The hypotenuse is OB.
• The radius OM=1 is the adjacent side.

cosβ = OM/OB

OB = 1/cosβ

ii) OA 

In the right triangle △OMA

The hypotenuse is OAOAOA.

The radius OM=1 is the opposite side.

Length of OA:

sinβ = OM/OA

OA = 1/sinβ

iii) AB

In the right triangle △OAB

∠AOB=90°

By the Pythagorean theorem: 

AB²=OA²+OB²

Substituting OA=1/sinβ and OB = 1/cosβ

AB²=(1/sinβ)²+(1/cosβ)²

AB²=(1/sin²β)+(1/cos²β)

AB²=(sin²β+cos²β)/(sin²β.cos²β)

Using the Pythagorean identity (sin²β+cos²β)= 1

AB²=1/(sin²β.cos²β)

AB=1/(sinβ.cosβ)

AB=2/(sin2β)

5. In the figure below, 5sin P = 4.

(Note: The figure is not to scale.)

What is the length of PR? Draw a diagram and show your steps.

Soln.

In △PQR

PQ=QR=15 units.

We need to find the length of PR.

5sin⁡P=4

sin⁡P= ⅘

In △PQR, the Law of Cosines states:

PR²=PQ²+QR²−2⋅PQ⋅QR⋅cosP.

Substitute PQ=QR=15

PR²=15²+15²−2⋅15⋅15⋅cosP.

Use the Pythagorean identity:

sin⁡²P+cos⁡²P=1.

Substitute sin⁡P= ⅘

(4./5)²+cos⁡²P=1.

(16/25) +cos⁡²P=1.

cos⁡²P = 1- (16/25)

cos⁡²P = 9/25)

cos⁡P = ± 3/5

Since P is acute, 

PR²=15²+15²−2⋅15⋅15⋅± ⅗

PR²=(225+225−2⋅15⋅15⋅3)/5

.PR²=180

.PR=6√5

6. Shown below is a cuboid. Its length is / units, breadth b units and height h units.

i) Express cos θ in terms of I, b, and h.

ii) If the figure was a cube, what would be the value of cos θ ?

Show your work.

Soln. 

Given:

The cuboid has:

• Length l,
• Breadth b,
• Height h, and θ is the angle between the diagonal AC and the base diagonal CG.

i) Express cos θ in terms of I, b, and h.

Calculate the length of CG:CG is the diagonal of the rectangular base ABCD.

CG = √l² + b²

Calculate the length of AC:AC  is the space diagonal of the cuboid

CG = √l² + b²+h²

 cos θ = CG/ AC

 cos θ = (√l² + b²+h²)/( √l² + b²⁾

ii) If the figure was a cube, what would be the value of cos θ 

Substitute the value of l, b, and h

 cos θ =(√l² + b²⁾/ (√l² + b²+h²)

In cube, every side is equal l=b=h

 cos θ =(√l² +l²⁾/ (√l² + l²+l²)

 cos θ =(√2l²/ (√3l²

 cos θ =(√2 l/ (√3) l

 cos θ =(√2 / (√3)

after rationalising

 cos θ =√6 / 3

7. The teacher asked the students to correctly complete the following sentence about the rhombus.

"A rhombus has a side length of I units and one of its angles is equal to 0. The ratio of the lengths of the two diagonals is dependent on

Ashima: only I.

Bilal: only θ.

Chris: both l and θ.

Duleep: neither l nor θ.

Who answered the question correctly? Show your work and give valid reasons.

Given that a rhombus has a side length of l units and one of its angles is equal to θ. We have to find the dependency of the ratio of the lengths of the two diagonals.

First, we draw a rhombus ABCD whose side length is l units, diagonals AC and BD intersect at E, and ∠ABC=θ as shown below-

We know that in a rhombus, all sides are equal and diagonals bisect each other at 90°, and diagonals bisect the angles of a rhombus. i.e.

∠BEC=∠CED=∠AED=∠AEB=90°,

AE=CE, BE=DE, and

∠ABD=∠CBD, ∠BCA=∠DCA, ∠BAC=∠DAC, ∠ADB=∠CDB

So, ∠ABD=∠CBD=θ/2 or ∠ABE=∠CBE=θ/2

The tangent (tan) of an angle in a right-angled triangle is defined as the ratio between the lengths of opposite side of the angle to its adjacent side.

Now, in right-angled ∆BEC-

tanθ/2=CE/BE=(1/2AC)/(1/2BD)=AC/BD

As we can see that the ratio of the diagonals depends only on θ.

Hence, Bilal answered correctly.

8. A 90 cm wide door opens on one side of the room at a maximum angle of 90°. Due to shortage of space, a 40 cm by 80 cm table is kept behind the door along the wall such that it obstructs its path.

(Note: The figure is not to scale.)

At what distance from the hinge should the table be kept such that the door opens for a maximum angle of 60°. Show your work.

(Note: Use √2 = 1.41, √3 = 1.73)

Soln. 

Given that a 90 cm wide door opens on one side of the room at a maximum angle of 90°. Due to shortage of space, a 40 cm by 80 cm table is kept behind the door along the wall such that it obstructs its path.

Let at a distance of x cm from the hinge should the table be kept such that the door opens for a maximum angle of 60° as shown below-

From the figure, ∠ACB=90°-60°=30°.

So, in right-angled triangle ABC-

tan30°=AB/AC =40/x

= 1/√3 =40/x

⇒x=40/√3

=40×1.73

=69.2 cm

Hence, 69.2 cm is the required answer.

What is Trigonometry?

Simply put, trigonometry is the study of triangles, particularly right-angled triangles, and the relationships between their angles and sides. The word comes from the Greek terms "trigonon" (triangle) and "metron" (measure), meaning it’s all about measuring triangles.

Trigonometry deals with ratios. When you take a right-angled triangle, there are three key trigonometric ratios you’ll need to learn:

• Sine (sin): The ratio of the length of the opposite side to the hypotenuse.
• Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan): The ratio of the length of the opposite side to the adjacent side.

These ratios help us figure out unknown sides and angles in triangles. In Class 10, you’ll learn how to use them in a variety of problems, from calculating the heights of objects to determining distances across rivers. While this might seem theoretical, these skills come in handy for solving real-world problems, especially in fields like engineering, physics, and architecture.

Why Should You Need Class 10 Trigonometry Important Questions? 

Here are a few reasons why mastering trigonometry is mandatory, especially in class 10.

• Trigonometry also serves as a foundation for several other complex mathematical subjects that you will study in the future, such as calculus and physics, among others. Understanding trigonometry in advance will make it easier to grasp in other educational classes.
• Trigonometry is not merely a subject a learner encounters during exams. Trigonometry finds its application in various fields such as architecture, cartography, and satellite distance determination. Trigonometry is even used in the design of video games to create 3D illustrations. The more you know it today, the more beneficial it will be tomorrow. 
• Solving trigonometry problems enhances an individual's analytical skills by facilitating the development of problem-solving steps. For this reason, all activities, whether in scientific, technological, social, or even physical aspects of human life, can benefit from such a logical approach. 
• Trigonometry is one of the most straightforward and high-scoring topics in your Class 10 board exams, provided you understand the fundamentals. Once you grasp the concepts of trigonometry well, they become quite understandable, and with ample practice, you can easily answer all these questions to achieve full marks in this portion.

How to Master Trigonometry Class 10 Important Questions?

Important questions in the introduction to trigonometry class 10 level require a smart and proper approach to master. Let’s go over a few key tips to help you become confident in solving trigonometric problems:

• Do not directly jump to solving problems; always make sure that you have a proper understanding of elementary trigonometric functions. Remember the Pythagorean theorem, the principles of the connection between angles and sides in right-angled triangles, and seek to memorize the trigonometric ratios for sine, cosine, and tangent. This will make it easier to apply good concepts in answering those questions. 
• Trigonometry commonly employs standard angles such as 30°, 45°, 60°, and 90°. This implies that the ratios of sin, cos, and tan for these angles should be familiar to all students.
• Trigonometric identities are formulas that simplify equations. For instance, the identity sin2θ + cos2θ = 1 is commonly applied to solve tough problems.  Revise the key identities and try to solve different questions using the various identities. 
• The fundamentals of trigonometry rely on the visualization of angles and sides. Whenever you are solving a question that forms the basis of your answer, always take out your pencil and create a neat and workable diagram. Make certain that the labels of the sides and angles are quite distinct. That is why looking at the definition of a special right triangle makes it easier to determine which trigonometric ratio to apply.
• Taking past papers is one of the excellent ways to prepare for your exams. Preparing mock tests relieves your anxiety about the examination pattern, and you'll likely encounter similar questions from the mock tests during the actual exam. Try to solve the questions from the textbook exercises, and also practice some additional easy and difficult questions related to trigonometry class 10, along with the answers from previous examinations.

Solving Extra Questions for Class 10 Maths Trigonometry not only prepares you for exams but also enhances your problem-solving skills in various areas. The best way to really understand trigonometry is to start with the basics and then work through the important questions of trigonometry class 10 frequently. 

Trigonometry is a subject that greatly benefits from practice, and it's widely known that mastering it not only in math but across all subjects can lead to high exam scores or enhance your analytical thinking skills. If you master the appropriate learning and practice strategies, this chapter provides a solid foundation for future mathematics lessons.