Write each expression as the function of an acute angle whose measure is less than 45 degrees:

Question

Write each expression as the function of an acute angle whose measure is less than 45 degrees:

Cos 110 degrees b) tan 265 degrees .

Answer 1

The "co" in "cosine" means "complement"

cos110° = -cos70° = -sin20°
tan265° = tan85° = cot5°
review angles in various quadrants.

Answer 2

a) To write cos 110 degrees as the function of an acute angle, we can use the fact that cosine is an even function. This means that cos(-x) = cos(x) for any angle x. Therefore, we can find an equivalent acute angle by subtracting 360 degrees.

cos 110 degrees = cos (-250 degrees).

Now, we want to find an acute angle whose measure is less than 45 degrees. We can achieve this by subtracting 360 degrees again.

cos (-250 degrees) = cos (110 - 360) degrees.

So, we can write cos 110 degrees as the function of an acute angle whose measure is less than 45 degrees as:

cos 110 degrees = cos (110 - 360) degrees.

b) To write tan 265 degrees as the function of an acute angle, we can use the fact that tangents have a periodicity of 180 degrees. That means that tan (x + 180 degrees) = tan(x) for any angle x. So, we can subtract 180 degrees to find an equivalent acute angle.

tan 265 degrees = tan (265 - 180) degrees.

Now, let's simplify it and find an acute angle whose measure is less than 45 degrees.

tan (265 - 180) degrees = tan 85 degrees.

Therefore, we can write tan 265 degrees as the function of an acute angle whose measure is less than 45 degrees as:

tan 265 degrees = tan 85 degrees.

Answer 3

To write each expression as the function of an acute angle less than 45 degrees, we can use the following trigonometric identities:

1. For cosine (cos) function: cos(180° - θ) = -cos(θ)
2. For tangent (tan) function: tan(θ + 180°) = tan(θ)

a) cos 110 degrees:
Since 110 degrees is greater than 90 degrees, we can subtract 90 degrees from it to get an acute angle:

110° - 90° = 20°

Now, we can use the identity cos(180° - θ) = -cos(θ) to rewrite it as:

cos 110° = -cos 20°

So, the function of an acute angle less than 45 degrees that has the same cosine value as cos 110° is -cos 20°.

b) tan 265 degrees:
To convert 265 degrees to an acute angle, we can subtract multiples of 180 degrees until we get an angle less than 45 degrees:

265° - 180° = 85°
85° - 180° = -95°
-95° + 180° = 85°

Now, we can use the identity tan(θ + 180°) = tan(θ) to rewrite it as:

tan 265° = tan 85°

So, the function of an acute angle less than 45 degrees that has the same tangent value as tan 265° is tan 85°.