CBSE Class 10 · Mathematics
Some Applications of Trigonometry
Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.
This lesson follows the official chapter “Some Applications of Trigonometry” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 10
- Medium level
- 8 concepts
1Heights and Distances
A height-distance story needs one right triangle: a vertical (or horizontal) you can name, a line of sight, and a right angle. The unknown is a side; the given is an angle plus one side. Two triangles glued without a named shared side are a second story.
A castle-photo with no right angle marked is not this story.
Figure. A height-distance question is one right triangle: the standing height, the ground distance, and the line of sight. The angle lives at the eye.
How it works
- Draw the vertical and the sight-lineThe triangle.
- Mark the right angle and the given angleThe marks.
- Name the unknown sideHeight or distance.
2A ratio of two sides of a right triangle
In that triangle the same ratios apply: opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent. Application means you pick the ratio that holds the given side and the unknown. A ratio of two unknown sides is a stuck write.
Using sin because the word “sine” was in the chapter title, when the given was adjacent and opposite, is a miss.
Figure. From the angle, opposite is the side that faces it, adjacent is the other leg, hypotenuse is the long side. A trig ratio is two of those lengths.
How it works
- Name given side and unknown side relative to the angleOpp, adj, hyp.
- Pick the ratio that holds bothsin, cos, or tan.
- Solve for the unknownThe application.
3Sine, cosine, and tangent name those ratios
Sine, cosine, and tangent still name those ratios here — now in a tower-or-string story, not a bare triangle drill. The names did not change because the story got a person in it.
Renaming tan as “the height button” is a heading-steal.
Figure. Sine, cosine, and tangent are only names for those three side-pairs. They do not add a fourth side.
How it works
- Keep sin, cos, tan as opp/hyp, adj/hyp, opp/adjThe names.
- Drop them onto the height-distance triangleThe same names.
- Refuse a new everyday nickname as the mathsThe ratios stay.
4Special angles have exact values you can store
Special angles still have exact values you can store: a 30° sight with opposite 10 m gives hypotenuse 20 m (sin 30° = 1/2). Application is the store plus the story, not a new 30°.
Replacing 1/2 by 0.3 because the tower “looked shorter” is a miss.
Figure. A 30-60-90 triangle stores the exact sides 1, √3, 2. Those sides give the exact sine, cosine, and tangent you memorise.
How it works
- See 30°, 45°, or 60°The special.
- Write the stored exact ratioThe store.
- Solve the side in the storyExact, then the length.
10 m opposite, 30°
A sight-angle 30° has opposite side 10 m. Find the hypotenuse.
- sin 30°1/2 = opp/hyp
- hyp10 / (1/2) = 20 m
- ReadStore plus the story
Pro tip. Special-angle store; the person in the story does not change 1/2.
5An identity is a rewrite that stays true
An identity can still rewrite a ratio (sin/cos = tan) so the given pair becomes the unknown. It stays true; it is not a second height. Use a rewrite only when it unlocks the side you have.
Expanding sin²+cos² in a story that already has tan and one side is a detour unless it helps.
Figure. On a hypotenuse of 1, the legs are sin θ and cos θ. Pythagoras is then the identity sin²θ + cos²θ = 1 — a rewrite that stays true for every allowed θ.
How it works
- See if a rewrite turns given into unknownThe identity-job.
- Keep it true for the angleThe rewrite.
- Then return to the sideApplication, not a new drill.
6A height-distance story needs one right triangle
The whole story is one right triangle unless the lesson glued two (a depression plus a base, as taught). Name that one triangle before you invent a second tower.
Two buildings in a photo are not two triangles until you mark them.
Figure. Draw only the triangle the story names: eye, foot of the tower, top of the tower. Extra buildings are a different question.
How it works
- Count the right triangles you actually markedUsually one.
- Put the given angle and side on that triangleThe story.
- Solve one unknown; a second only if a second triangle was markedEnough.
7The angle of elevation sits at the eye
The angle of elevation sits at the eye: from the horizontal up to the sight-line. It is not the angle at the tower-foot unless those two happen to be equal by a parallel-lines story you argued. Elevation is a place-name for the angle.
Putting 30° at the foot “because heights live there” is a silent move.
Figure. Angle of elevation is the up-angle between the horizontal through the eye and the line of sight. It is not at the tower foot.
How it works
- Draw the horizontal at the eyeThe level.
- Draw the sight-line upThe look.
- Mark elevation between those twoAt the eye.
8Depression is the down-angle from the horizontal
Depression is the down-angle from the horizontal at the eye (or at a high stand). A parallel-lines argument can move it to an equal angle at the base if the lesson did that. Depression is not “elevation with a frown” unless that equality is shown.
Using depression as if it already sat at the foot with no parallel is a miss.
Figure. Depression is the down-angle from the horizontal at the eye. The horizontal is still at eye height — not along the ground.
How it works
- Draw the horizontal at the high eyeThe level.
- Draw the sight-line downThe look.
- Mark depression there; move it only with a taught parallelThe down-angle.
Angle of elevation sits
- At the eye, between horizontal and the up-sight
- Always at the tower foot
- At the midpoint
Place-name for the angle.
Notes
- The official chapter title is “Some Applications of Trigonometry”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- elevation at the eye, up from horizontal
- depression down from horizontal; parallel may move it
Recap
Hold these pegs from the official chapter “Some Applications of Trigonometry”. The wording is ExamMaster’s teaching, not a textbook recap.
- Heights and Distances
- A height-distance story needs one right triangle: a vertical (or horizontal) you can name, a line of sight, and a right angle.
- A ratio of two sides of a right triangle
- In that triangle the same ratios apply: opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent.
- Sine, cosine, and tangent name those ratios
- Sine, cosine, and tangent still name those ratios here — now in a tower-or-string story, not a bare triangle drill.
- Special angles have exact values you can store
- Special angles still have exact values you can store: a 30° sight with opposite 10 m gives hypotenuse 20 m (sin 30° = 1/2).
- An identity is a rewrite that stays true
- An identity can still rewrite a ratio (sin/cos = tan) so the given pair becomes the unknown.
- A height-distance story needs one right triangle
- The whole story is one right triangle unless the lesson glued two (a depression plus a base, as taught).
Practise Some Applications of Trigonometry
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device