Solve this question---**Diagram Description:**
* **Type:** Geometric figure (Triangle).
* **Main Elements:**
* A triangle labeled ABC.
* A line segment BD connects vertex B to a point D on the side AC.
* The angle at vertex A (∠BAC) is labeled as 34°.
* The angle formed by segment BD and side AC at point D is a right angle (90°), indicated by a square symbol. This implies BD is perpendicular to AC.
* Vertices are labeled A, B, and C. Point D is on the line segment AC.
**Question Text:**
In triangle ABC above, AB = BC. what is the value of sin(∠CBD) - cos(∠BAC) ?
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We have triangle ABC where AB equals BC, making it an isosceles triangle. Angle BAC is 34 degrees, and BD is perpendicular to AC. We need to find the value of sine of angle CBD minus cosine of angle BAC.
Since AB equals BC, triangle ABC is isosceles. In an isosceles triangle, the base angles are equal. This means angle BAC equals angle BCA. Since angle BAC is 34 degrees, angle BCA is also 34 degrees.
Now we focus on triangle BDC. Since BD is perpendicular to AC, angle BDC is 90 degrees. Using the fact that angles in a triangle sum to 180 degrees, we have angle CBD plus angle BDC plus angle BCD equals 180 degrees. Substituting the known values: angle CBD plus 90 degrees plus 34 degrees equals 180 degrees. Solving for angle CBD: it equals 180 minus 90 minus 34, which is 56 degrees.
Now we evaluate the expression sine of angle CBD minus cosine of angle BAC. Substituting our values, we get sine of 56 degrees minus cosine of 34 degrees. We can use the complementary angle identity: cosine of x equals sine of 90 degrees minus x. Applying this to cosine of 34 degrees, we get sine of 56 degrees. So our expression becomes sine of 56 degrees minus sine of 56 degrees, which equals zero.
To summarize our solution: We identified that triangle ABC is isosceles with equal base angles of 34 degrees each. Using the angle sum property in triangle BDC, we found angle CBD equals 56 degrees. By applying the complementary angle identity, cosine of 34 degrees equals sine of 56 degrees. Therefore, sine of angle CBD minus cosine of angle BAC equals zero.