આ MCQ મોડ્યુલ આના પર આધારિત છે: Trigonometric Ratios of Specific Angles
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આ મૂલ્યાંકન આના પર આધારિત હશે: Trigonometric Ratios of Specific Angles Class 10 સ્તર, Trigonometry માં, Intermediate કઠિનતા સાથે લક્ષ્ય.
મૂલ્યાંકન બનાવવામાં તેમની સામગ્રી સામેલ કરવા ચિત્રો, PDF અથવા Word દસ્તાવેજ અપલોડ કરો.
8.3 Trigonometric Ratios of Some Specific Angles
From geometry you already know shapes like the isosceles right trianglei (45°-45°-90°) and the equilateral trianglei (60°-60°-60°). We now compute the trigonometric ratios of 0°, 30°, 45°, 60° and 90°.
Ratios of 45°
Consider an isosceles right triangle ABC, right-angled at B, with \(AB=BC=a\). Then \(\angle A=\angle C=45°\) and by Pythagoras \(AC=a\sqrt2\).
Fig. 8.14 — Isosceles right triangle for 45° ratios
Consider an equilateral triangle ABD with each side \(2a\). Drop the perpendicular AC from A to BD. Then BC = CD = \(a\), AC = \(a\sqrt3\), and \(\angle BAC=30°,\;\angle ABC=60°\).
Fig. 8.15 — Half of an equilateral triangle gives 30°–60°–90°
When angle A is made smaller and smaller in a right triangle (with BC held fixed shrinking, or with angle approaching the base), side BC approaches 0 and AC approaches AB. Taking limits:
\[\sin 0°=0,\;\cos 0°=1,\;\tan 0°=0,\;\sec 0°=1,\;\csc 0°\text{ and }\cot 0°\text{ are not defined.}\]
When A is close to 90°, AB approaches 0 and BC approaches AC:
\[\sin 90°=1,\;\cos 90°=0,\;\tan 90°\text{ and }\sec 90°\text{ are not defined,}\;\csc 90°=1,\;\cot 90°=0.\]
Table 8.1 — Standard Values
∠A
0°
30°
45°
60°
90°
sin A
0
1/2
1/√2
√3/2
1
cos A
1
√3/2
1/√2
1/2
0
tan A
0
1/√3
1
√3
Not defined
cosec A
Not defined
2
√2
2/√3
1
sec A
1
2/√3
√2
2
Not defined
cot A
Not defined
√3
1
1/√3
0
Remark. As A increases from 0° to 90°, sin A increases from 0 to 1, while cos A decreases from 1 to 0.
Activity — Memorise the Table with Fingers
Trick: Hold up your left hand. Number fingers 0–4 for 0°, 30°, 45°, 60°, 90°.
For sin, count how many fingers are below the chosen angle, take square root, divide by 2.
For cos, count fingers above the chosen angle, square-root, divide by 2.
Try sin 45°: two fingers below. √2/2 = 1/√2. ✓
sin: 0, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1. cos: reverse. A quick mnemonic that students never forget.
Example 6
In \(\triangle ABC\) right-angled at B, AB = 5 cm and \(\angle ACB=30°\). Find BC and AC.
Solution. AB is opposite C. So \(\tan C=\dfrac{AB}{BC}\Rightarrow \tan 30°=\dfrac{5}{BC}\Rightarrow \dfrac{1}{\sqrt3}=\dfrac{5}{BC}\Rightarrow BC=5\sqrt3\) cm.
Also \(\sin 30°=\dfrac{AB}{AC}\Rightarrow \dfrac{1}{2}=\dfrac{5}{AC}\Rightarrow AC=10\) cm. Alternatively \(AC=\sqrt{25+75}=10\) cm.
Example 7
In \(\triangle PQR\) right-angled at Q, PQ = 3 cm, PR = 6 cm. Determine \(\angle QPR\) and \(\angle PRQ\).
A: sin 30° + cos 60° = 1. R: sin 30° = cos 60° = 1/2.
A) Both true; R explains A
B) Both true; R does not explain A
C) A true, R false
D) A false, R true
Answer: A. Each equals 1/2, sum 1. R gives the values needed to compute A.
A: tan 90° is not defined. R: At 90°, the adjacent side vanishes and division by zero is undefined.
A) Both true; R explains A
B) Both true; R does not explain A
C) A true, R false
D) A false, R true
Answer: A. tan = opposite/adjacent; the adjacent side has length 0 at 90°.
Key Term
Frequently Asked Questions
Why is sin 30 degrees equal to 1/2?
In a 30-60-90 right triangle, the side opposite the 30-degree angle is half the hypotenuse. So sin 30 = opposite/hypotenuse = 1/2.
Why are sin 45 and cos 45 equal?
In a 45-45-90 right triangle, the two legs are equal and the hypotenuse is root 2 times a leg. Both sine and cosine equal leg/hypotenuse = 1/root 2.
What are the values at 0 and 90 degrees?
sin 0 = 0, cos 0 = 1, tan 0 = 0. sin 90 = 1, cos 90 = 0, tan 90 is undefined because cos 90 is 0.
Why is tan 90 undefined?
Because tan A = sin A / cos A, and cos 90 = 0 makes the denominator zero, so tan 90 is undefined (approaches infinity).
How do these values help in problems?
They allow exact computation in geometry, physics and engineering without a calculator, especially for problems involving 30-60-90 or 45-45-90 right triangles.
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