Most Important Questions Class 10 Maths Some Applications of Trigonometry with Solutions
Mathematics is challenging for many students, but understanding its basics can improve your mathematics. One such topic in Class 10 is trigonometry. Trigonometry is essential not just for class 10 but also for competitive exams like JEE. We know you will be thinking about why practicing trigonometry important questions for Class 10 is important and how these questions can improve your confidence. Well, we will provide answers to all your questions along with Class 10 Applications of Trigonometry Important Questions.
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Chapter 9 Some Applications of Trigonometry: Important Questions
1. Shown below is a rectangular tub of water of depth 34 cm. An object O is at the bottom of the tub. The image of the object is formed at I for an observer at Q.
(Note: The figure is not to scale.)
Find the distance by which the object seem to be moved for the observer. Show your work and give valid reasons.
(Note: Take √2 = 1.4, √3 = 1.7)
Soln.
Real depth of the object: d=34 cm
Angle of incidence (i) = 45°
Angle of refraction (r) = 30°
Snell's Law: n1sini=n2sinr
Here, n1=1(air refractive index) and n2=μ (water refractive index).
sin45°=μsin30°
Substituting values
0.7=μ×0.5
0.7/0.5=μ
1.4 =μ
apparent depth da= Real depth/μ
apparent depth da= 34/1.4
≈24.3cm
The object appears to move by the difference between the real depth and apparent depth:
Shift=d−da
Shift=34−24.3
=9.7cm.
The object seems to move upward by approximately 9.7 cm for the observer.
2. Answer the questions based on the information given below.
At an archery academy, Guru Drona had floated a gift box with two balloons at a height of H metres from the table. As part of his practice, Arjuna was given the task to bring the gift box to the table placed below. Arjuna was standing on the ground at a horizontal distance of 100 metres from the table at point B. He aimed at the balloons with an elevation angle of 9 and shot the arrow to burst one of the balloons.When Arjuna burst the first balloon, the box came down to the height of h metres from the table. He now reduced his angle of elevation by ẞ and shot his arrow at the second balloon. The second balloon burst and the gift box landed safely on the table. Assume that Arjuna's arrows travelled in straight lines and did not curve down.
(Use √3 = 1.73, √2 = 1.41)
2.1 If θ = 45° and ẞ = 15°, what is the difference between the box's initial height and its height after the first shot?
a. 100- 100 √3m
b. 100 √3m
c. 100√3 - 100 m
d. (cannot be calculated without knowing H.)
Soln. (c) 100√3 - 100 m
Explanation:
height H of the first balloon:
H=100tan45°
=100.
height h after the first shot:
h=100tan15°
h=100(2−√3)
Δh =H−h
=100−100(2−√3)
Δh=100√3−100
2.2 If θ = 45° and ẞ = 15°, what is the distance that the arrow has to travel to burst the second balloon?
a. 100√3/2 m
b. 200/√3 m
c. 100√2 m
d. 100 √3m
Soln. (d) 100 √3m
Explanation:
For the second balloon, at a height h:
h=100tan15°
100(2−√3)
slant distance d: h/sin15°
sin15°= (√6 - √2)/4
d= (100(2−√3))/((√6 - √2)/4)
d= 100 √3m
2.3 For Ashwatthama, Guru Drona raised the gift box further higher such that the angles θ and ẞ were 60° and 30° respectively. What is the value of the ratio H/h now?
a. 1/√3
b. √3
c. 2
d. 3
Soln. (d) 3
Explanation:
H=100tan60°
=100 √3
h=100tan30°
=100/√3
H/h = 100 √3/100/√3
= 3
2.4 When the initial angle of elevation, θ, was 45°, Arjuna felt uncomfortable as it strained his neck. From his original spot, approximately how much should he retreat away from the balloons, so that the new angle of elevation, θ, becomes 30°?
a. 73 m
b. 100 m
c. 173 m
d. (cannot be calculated without knowing H.)
Soln. (a) 73m
Explanation:
H=100tan45°
=100
the horizontal distance x is = H/tan 30°
x= 100/1√3
x= 100√3
Retreat=x−100
100√3 - 100
√3 ≈1.73
Retreat=173−100
=73m.
2.5 Arjuna measured that θ = 45°. Right before he could shoot the first arrow, a gust of wind pushed the balloons 15 m higher. What should Arjuna do to ensure that he doesn't miss?
a. Move towards the table by 15 m but keep the arrow at the same angle of elevation.
b. Move away from the table by 15 m but keep the arrow at the same angle of elevation.
c. Increase the arrow's angle of elevation by 15° but stay at the same place.
d. Move away from the table by 15 m and increase the arrow's angle of elevation by 15°.
Soln. (a) Move towards the table by 15 m but keep the arrow at the same angle of elevation.
Explanation:
With a rise of 15 m, the new height becomes H=115
To maintain the same angle of elevation θ=45° the horizontal distance must equal the new height (x=115)
Arjuna must move towards the table by.
Δx=115−100
=15m
Move towards the table by 15 m but keep the arrow at the same angle of elevation.
3. The position of an eagle and two identical geese are shown in the figure below. All the [ birds are at the same height from the ground. Assume that the Eagle can fly at the same speed in all directions and that the geese are unaware of the Eagle's intention and will not move from their positions.
(Note: The figure is not to scale.)
If the eagle wants to attack the goose that is nearer to it, which one should it attack? Show your steps.
(Note: Use √2 = 1.41, √3 = 1.73)
Soln.
The eagle is located at the origin of the diagram.
The positions of the geese are:
Goose 1: 100 m away at an angle of 45°.
Goose 2: 75 m away at an angle of 30°.
Note: The distances given are along the hypotenuse of their respective right triangles
Distance to Goose 1
d1= √(100. cos(45°))2 + (100. sin(45°))2
cos(45°)=sin(45°)= √2/2≈0.707
d1= √(100. 0.707)2 + (100. 0.707)2
d1= √(70.7)2 + (70.7)2
d1= √2 x (70.7)2
d1= √2 x 5000
d1= √10000
d1= 100
Distance to Goose 2
d2= √(75. cos(30°))2 + (75. sin(30°))2
d2= √(75. 0.866)2 + (75. 0.5)2
d2= √(64.95)2 + (37.5)2
d2= √4220.9 + 1406.252
d2= √5627.15
≈75m
Since d2=75 is shorter than d1=100 , the eagle should attack Goose 2, as it is closer.
4. In the giant wheel shown below, Gagan is sitting in one of the cabins which is 12 m high from the platform. Jyoti and Karan are sitting in the lowest and the highest cabins from the platform respectively.
(Note: The figure is not to scale.)
From Gagan, the angle of depression of Jothi and the angle of elevation of Kiran is 30° and 60° respectively.
i) What will be the angle of elevation of Gagan from Jothi?
ii) What will be the angle of depression of Gagan from Kiran?
iii) Find the diameter of the giant wheel.
Show your steps with a diagram.
Soln.
Let the height of the center of the wheel from the platform be h
Let the radius of the wheel be r, so h=r.
Gagan’s cabin is at 12 m, so h−12=r
Jyoti is at the bottom of the wheel, at height 0
Karan is at the top of the wheel, at height 2r
The angle of elevation from Jyoti to Gagan is 30°
Hence, the angle is complementary and equal to 30°
Gagan to Jyoti: From the geometry and the given angle of depression 30°, the relationship between the height h=12 and the radius r can be calculated.
Radius of the wheel:
From Jyoti to Gagan, the radius of the wheel is approximately 20.78 m
Diameter of the wheel:
The diameter of the giant wheel is 41.57 m
Height of Karan:
The height of Karan (topmost cabin) is approximately 48.0 m
The angle of elevation from Jyoti to Gagan is the same as the angle of depression from Gagan to Jyoti, which is 30°.
The angle of depression of Gagan from Karan is the same as the angle of elevation of Karan from Gagan, which is 60°.
The diameter of the wheel is 41.57 m
5. A ship was moving towards the shore at a uniform speed of 36 km/h. Initially, the ship was 1.3 km away from the foot of a lighthouse which is 173.2 m in height.
Find the angle of depression, x, of the top of the lighthouse from the ship after the ship had been moving for 2 minutes.
Show your steps and give reasons.
(Note: Take √3 as 1.732 and √2 as 1.414.)
Soln.
Height of the lighthouse = 173.2 m
Initial horizontal distance of the ship from the lighthouse = 1.3 km
Speed of the ship = 36 km
Time of travel = 2 minutes = 2/60 hours= 1/30hours
Distance = Speed ×Time
Distance traveled=36× 1/30
=1.2km.
The new horizontal distance = Initial distance −-− Distance traveled:
New distance=1.3−1.2
=0.1km.
tan(x)= Height of lighthouse/ New horizontal distance
= 0.1732/ 0.1
tan(x)=1.732
Since tan(x)=1.732, and we are given that √3=1.732,
x=60°
Overview of Introduction to Trigonometry Class 10
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right-angled triangles. Class 10 introduces students to fundamental trigonometric concepts like sine, cosine, and tangent. Understanding these concepts is important for handling problems in trigonometry and other topics like mensuration, calculus, and even physics.
But why should you focus on the 10 important questions in trigonometry? What’s the best way to make use of these problems to improve your performance? Let’s see the benefits:
Why should you use Trigonometry Important Questions for Class 10?
Students learn essential mathematical principles in trigonometry, which are applicable in both academic and real-world contexts. By practicing trigonometry with important questions for Class 10, students understand how trigonometric functions like sine, cosine, and tangent work.
Every student wants to score high marks in their Class 10 board exams, and practicing imp questions of trigonometry in Class 10 is an easy way to ensure success. These questions often represent the most important parts of the syllabus and frequently appear in exam papers. Solving them will make you familiar with the pattern of questions.
Trigonometry, like most areas of mathematics, is about problem-solving. The more you practice answering important questions from the introduction to trigonometry class 10, the more proficient you become in critical and analytical thinking.
Regularly practicing trigonometry's important questions for class 10 strengthens your understanding of the subject, which we see in improved performance in exams. When you can confidently answer difficult trigonometry questions, you’ll approach your exams with a positive mindset, reducing anxiety and stress.
Many students struggle with time management during exams, especially in subjects like mathematics. Practicing Imp questions in trigonometry class 10 helps students get familiar with the types of questions that are most likely to appear. This familiarity allows you to manage your time efficiently during exams, ensuring that you can complete all sections within the allocated time.
How to Make the Best Use of Trigonometry Class 10 Important Questions?
Now that we understand the importance of practicing trigonometry in class with 10 important questions, let’s learn how you can use them effectively to maximize your learning.
Before jumping into complex problems, it’s essential to ensure you understand the basics. Revise key trigonometric formulas regularly.
The trigonometry class's 10 important questions come in various types, ranging from straightforward computation problems to word problems that require a deeper understanding of concepts. Make sure to practice a mix of multiple-choice questions, short-answer questions, and long-answer questions. Each type presents a unique challenge, and solving them improves your overall problem-solving ability.
One of the best ways to get a feel for the types of questions that might appear in your board exam is to solve previous years’ papers. These papers will likely contain several trigonometry-important questions for class 10, giving you a view of frequently asked questions. The more you practice with these papers, the more confident and prepared you’ll feel.
It’s natural to make mistakes while practicing math problems, especially when it comes to trigonometry. However, rather than allowing errors to discourage you, embrace them as opportunities for learning. When you encounter a mistake, don’t simply move on; revisit the problem, understand why you got it wrong, and make sure you don’t repeat the same error.
Trigonometry involves many formulas, and memorizing them is crucial. Creating a personal formula chart and hanging it near your study space can greatly aid in memory retention.
Focusing on Applications of Trigonometry Extra Questions is a way to understand this essential area of mathematics. Whether you're solving basic problems or handling complex questions, consistent practice and strategic learning methods can help you achieve your dreams.
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