TRIGONOMETRIC RATIOS OF COMPLEMENTARY ANGLES
About "Trigonometric ratios of complementary angles"
Trigonometric ratios of complementary angles :
Two acute angles are complementary to each other if their sum is equal to 90°. In a right triangle the sum of the two acute angles is equal to 90°. So, the two acute angles of a right triangle are always complementary to each other.
Let ABC be a right triangle, right angled at B.
If <ACB = θ, then <BAC = 90° - θ and hence the angles <BAC and <ACB are complementary
For the angle θ, we have
Similarly, for the angle (90° - θ), we have
Comparing the equations in (1) and (2) we get,
Key concept - Trigonometric Ratios of Complementary Angles
Trigonometric ratios of complementary angles - Practice problems
Problem 1 :
Evaluate : cos 56° / sin 34°
Solution :
The angles 56° and 34° are complementary.
So, using trigonometric ratios of complementary angles, we have
cos 56° = cos (90° - 56°) = sin 34°
cos 56° / sin 34° = sin 34° / sin 34° = 1
Hence the value of cos 56° / sin 34° is 1.
Problem 2 :
Evaluate : tan 25° / cot 65°
Solution :
The angles 25° and 65° are complementary.
So, using trigonometric ratios of complementary angles, we have
tan 25° = tan (90° - 65°) = cot 65°
tan 25° / cot 65° = cot 65° / cot 65° = 1
Hence the value of tan 25° / cot 65° is 1.
Problem 3 :
Evaluate : (cos 65° sin 18° cos 58°) / (cos 72° sin 25° sin 32°)
Solution :
Using trigonometric ratios of complementary-angles, we have
cos 65° = cos (90° - 25°) = sin 25°
sin 18° = sin (90° - 72°) = cos 72°
cos 58° = cos (90° - 32°) = sin 32°
(cos 65° sin 18° cos 58°) / (cos 72° sin 25° sin 32°) is
= (sin 25° cos 72° sin 32°) / (cos 72° sin 25° sin 32°)
= 1
Hence the value of the given trigonometric expression is 1.
Problem 4 :
Prove : tan 35° tan 60° tan 55° tan 30° = 1
Solution :
Using trigonometric ratios of complementary-angles, we have
tan 35° = tan (90° - 55°) = cot 55° = 1/tan 55°
tan 60° = tan (90° - 30°) = cot 30° = 1/tan 30°
tan 35° tan 60° tan 55° tan 30° is
= (1/tan 55° ) x (1/tan 30°) tan 55° tan 30°
= 1
Hence, tan 35° tan 60° tan 55° tan 30° = 1
Problem 5 :
If sin A = cos 33°, find A
Solution :
Using trigonometric ratios of complementary-angles, we have
sin A = cos (90° - A)
Therefore,
sin A = cos 33° -----> cos (90° - A) = cos 33°
90° - A = 33°
90° - 33° = A
57° = A
Hence, A is 57°.
After having gone through the stuff given above, we hope that the students would have understood "Trigonometric ratios of complementary-angles"
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