please help me.
use trig. identities to find the exact value.
tan 25° + tan 5° / 1- tan 25° tan 5°
To find the exact value of the given expression, we can use the trigonometric identity for the sum of two tangents:
tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
Let's use this identity to simplify the expression:
tan 25° + tan 5° / (1 - tan 25° tan 5°)
First, let's calculate the individual tangent values:
tan 25° ≈ 0.4663
tan 5° ≈ 0.0875
Substituting these values into the expression:
tan 25° + tan 5° / (1 - (0.4663)(0.0875))
tan 25° + tan 5° / (1 - 0.04075)
tan 25° + tan 5° / 0.95925
To simplify further, we can rewrite tan 25° and tan 5° as ratios involving sine and cosine:
tan 25° = sin 25° / cos 25°
tan 5° = sin 5° / cos 5°
Substituting these values back into the expression:
(sin 25° / cos 25°) + (sin 5° / cos 5°) / 0.95925
Now, let's simplify further using the trigonometric identity:
sin(A) / cos(A) = 1 / tan(A)
Using this identity:
(1 / tan 25°) + (1 / tan 5°) / 0.95925
Now, we need to find 1 / tan 25° and 1 / tan 5°:
1 / tan 25° ≈ 2.1445
1 / tan 5° ≈ 11.4305
Substituting these values back into the expression:
2.1445 + 11.4305 / 0.95925
To simplify further, let's add the fractions:
(2.1445 + 11.4305) / 0.95925
13.575 / 0.95925
Lastly, we divide:
≈ 14.1233
Therefore, the exact value of the expression tan 25° + tan 5° / (1 - tan 25° tan 5°) is approximately 14.1233.
To find the exact value of the expression tan 25° + tan 5° / 1 - tan 25° tan 5° using trigonometric identities, we can use the tangent sum identity. The tangent sum identity states that the tangent of the sum of two angles is equal to the sum of the tangents of the individual angles divided by 1 minus the product of the tangents of the individual angles.
So, we can rewrite the expression as (tan 25° + tan 5°) / (1 - tan 25° tan 5°).
Now, we need to find the exact values of tan 25° and tan 5°.
First, let's find the value of tan 25°. We can use the tangent function and a calculator or trigonometric tables to find its approximate value. The approximate value of tan 25° is 0.4663.
Next, let's find the value of tan 5°. Similarly, we can use the tangent function and a calculator or trigonometric tables to find its approximate value. The approximate value of tan 5° is 0.0875.
Now, substitute these approximate values into the expression:
(tan 25° + tan 5°) / (1 - tan 25° tan 5°)
= (0.4663 + 0.0875) / (1 - 0.4663 * 0.0875)
= 0.5538 / 0.9584
≈ 0.5770
Therefore, the exact value of the expression tan 25° + tan 5° / 1 - tan 25° tan 5° is approximately 0.5770.
tan(25+5)= tan25+5/1-tan25tan5
tan(30)= tan25+5/1-tan25tan5
1/root(3)= tan25+5/1-tan25tan5
ur answer is 1/root(3)