Multiple Choice
Which trigonometric function is typically used to find the angle of elevation from a point on the ground to the top of a building when the height of the building and the horizontal distance from the building are known?
2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
Multiple Choice
In a right triangle, if angles and are the two non-right angles, what is the value of the product and the product ?
A
Both products equal
B
The first product is and the second is
C
Both products equal
D
The first product is and the second is
Verified step by step guidance1
Recall that in a right triangle, the two non-right angles \( b \) and \( c \) satisfy the relationship \( b + c = 90^\circ \) or \( b + c = \frac{\pi}{2} \) radians.
Use the complementary angle identity for sine and cosine: since \( c = 90^\circ - b \), we have \( \sin(c) = \sin(90^\circ - b) = \cos(b) \) and similarly \( \sin(b) = \cos(c) \).
Express the tangent functions in terms of sine and cosine: \( \tan(c) = \frac{\sin(c)}{\cos(c)} \) and \( \tan(b) = \frac{\sin(b)}{\cos(b)} \).
Substitute these into the products:
First product: \( \sin(b) \times \tan(c) = \sin(b) \times \frac{\sin(c)}{\cos(c)} \).
Second product: \( \sin(c) \times \tan(b) = \sin(c) \times \frac{\sin(b)}{\cos(b)} \).
Replace \( \sin(b) \) and \( \sin(c) \) using the complementary angle identities and simplify the expressions to show that both products equal 1.
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