Questions: Trigonometric Functions Relationship between the sines and cosines of complementary angles Complete the following. Part 1: Use t, u, and v to fill in the blanks. Make sure to use the appropriate upper-case or lower-case letters. sin T= / cos T= / cos U= / sin U= / Part 2: In triangle TUV, angle T and angle U are (Choose one) Part 3: Select all of the true statements. sin T=sin U cos T=sin U sin T=cos U cos T=cos U None of the above is true. Part 4: Fill in the blank. cos (34°)=sin ( °)

Trigonometric Functions
Relationship between the sines and cosines of complementary angles

Complete the following.

Part 1: Use t, u, and v to fill in the blanks.
Make sure to use the appropriate upper-case or lower-case letters.
sin T= /  cos T= / 
cos U= /  sin U= / 

Part 2: In triangle TUV, angle T and angle U are  (Choose one)
Part 3: Select all of the true statements.
sin T=sin U
cos T=sin U
sin T=cos U
cos T=cos U
None of the above is true.
Part 4: Fill in the blank.
cos (34°)=sin ( °)
Transcript text: Trigonometric Functions Relationship between the sines and cosines of complementary angles Complete the following. Part 1: Use $t, u$, and $v$ to fill in the blanks. Make sure to use the appropriate upper-case or lower-case letters. \[ \begin{array}{l} \sin T=\frac{\square}{\square} \quad \cos T=\frac{\square}{\square} \\ \cos U=\frac{\square}{\square} \quad \sin U=\frac{\square}{\square} \end{array} \] Part 2: In $\triangle T U V, \angle T$ and $\angle U$ are $\square$ (Choose one) Part 3: Select all of the true statements. $\sin T=\sin U$ $\cos T=\sin U$ $\sin T=\cos U$ $\cos T=\cos U$ None of the above is true. Part 4: Fill in the blank. $\cos \left(34^{\circ}\right)=\sin \left(\square^{\circ}\right)$
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Solution

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Solution Steps

Step 1: Determine the trigonometric ratios for angle T
  • sin T is the ratio of the opposite side to the hypotenuse.
  • cos T is the ratio of the adjacent side to the hypotenuse.

Given the triangle:

  • Opposite side to angle T: \( u \)
  • Adjacent side to angle T: \( t \)
  • Hypotenuse: \( v \)

Thus: \[ \sin T = \frac{u}{v} \] \[ \cos T = \frac{t}{v} \]

Step 2: Determine the trigonometric ratios for angle U
  • cos U is the ratio of the adjacent side to the hypotenuse.
  • sin U is the ratio of the opposite side to the hypotenuse.

Given the triangle:

  • Adjacent side to angle U: \( u \)
  • Opposite side to angle U: \( t \)
  • Hypotenuse: \( v \)

Thus: \[ \cos U = \frac{u}{v} \] \[ \sin U = \frac{t}{v} \]

Step 3: Identify the relationship between angles T and U

In \(\triangle TUV\), angles \( \angle T \) and \( \angle U \) are complementary. This means: \[ \angle T + \angle U = 90^\circ \]

Final Answer

  1. \(\sin T = \frac{u}{v}\)
  2. \(\cos T = \frac{t}{v}\)
  3. \(\cos U = \frac{u}{v}\)
  4. \(\sin U = \frac{t}{v}\)
  5. In \(\triangle TUV\), \(\angle T\) and \(\angle U\) are complementary.
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