Use fundamental identities and or the complementary angle theorem to find the value of the expression: tan 55°-cos 35°/sin 35°
a.) sqrt2/2
b.) sqrt3
c.) sqrt3/2
d.) sqrt2
e.) 0
f.) 1
notice that 55° and 35° are complementary angles, so
tan 55°-cos 35°/sin 35°
= tan55° - cot35°
= cot35° - cot35°
= 0
To find the value of the expression tan 55° - cos 35°/sin 35°, we can start by simplifying each term individually using the fundamental trigonometric identities.
1. Simplifying tan 55°:
The tangent function is defined as the ratio of the sine and cosine functions: tan θ = sin θ / cos θ.
Using this identity, we can rewrite tan 55° as sin 55° / cos 55°.
2. Simplifying cos 35°:
The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right angle triangle: cos θ = adjacent / hypotenuse.
Unfortunately, since there is no specific triangle given, we cannot directly use this definition to find the value of cos 35°. However, we can use the complementary angle theorem, which states that the cosine of an angle is equal to the sine of its complementary angle: cos (90° - θ) = sin θ.
So, we can rewrite cos 35° as sin (90° - 35°) = sin 55°.
3. Simplifying sin 35°:
We already know the value of sin 35° from the step above, which is sin 35° = sin (90° - 55°) = cos 55°.
Now, let's substitute the simplified terms back into the expression: (sin 55° / cos 55°) - cos 35° / sin 35°.
(sin 55° / cos 55°) - (cos 55° / sin 55°)
Next, we can rationalize the denominator of the second term by multiplying both the numerator and denominator by sin 55°:
(sin 55° / cos 55°) - (cos 55° * sin 55° / sin 55° * sin 55°)
(sin 55° / cos 55°) - (cos 55° * sin 55° / sin^2 55°)
Now, combining the two terms with a common denominator:
(sin 55° - cos 55° * sin 55°) / cos 55° * sin^2 55°
Factoring out the common factor of sin 55°:
sin 55° * (1 - cos 55°) / cos 55° * sin^2 55°
Using the identity sin^2 θ = 1 - cos^2 θ:
sin 55° * (1 - cos 55°) / cos 55° * (1 - cos^2 55°)
sin 55° * (1 - cos 55°) / cos 55° * (1 - cos 55°)
Simplifying further:
sin 55° / cos 55°
Finally, we can use the definition of the tangent function, tan θ = sin θ / cos θ, to get:
tan 55°
Therefore, the value of the expression tan 55° - cos 35°/sin 35° is equal to tan 55°.
Now, using a calculator or a trigonometric table, we can find the approximate value of tan 55°, which is approximately 1.4281.
So, the answer is f.) 1.
To find the value of the expression tan 55° - cos 35° / sin 35°, we will use the fundamental identity involving tan and sin.
The fundamental identity is: tan(x) = sin(x) / cos(x)
Using this identity, we can rewrite the expression as:
tan 55° - cos 35° / sin 35°
= sin 55° / cos 55° - cos 35° / sin 35°
Now, let's find the value of each trigonometric function:
sin 55° ≈ 0.8192
cos 55° ≈ 0.5736
cos 35° ≈ 0.8192
sin 35° ≈ 0.5736
Substituting these values into the expression, we get:
0.8192 / 0.5736 - 0.8192 / 0.5736
Now, we can combine the fractions by finding the common denominator:
(0.8192 * 0.5736 - 0.8192 * 0.5736) / 0.5736
Simplifying, we have:
(0.4709 - 0.4709) / 0.5736
= 0 / 0.5736
= 0
Therefore, the value of the expression tan 55° - cos 35° / sin 35° is 0, which corresponds to option e.) 0.