Find the remaining trigonometric ratios of 𝜃 based on the given information.

sin 𝜃 = −

3

2

and 𝜃 is not in QIII

cos 𝜃

=

0.745

tan 𝜃

=

−0.894

cot 𝜃

=

−1.118

sec 𝜃

=

1.357

csc 𝜃

=

Given the following trigonometric ratios:

- \( \sin \theta = -\frac{3}{2} \)
- \( \cos \theta = 0.745 \)
- \( \tan \theta = -0.894 \)
- \( \cot \theta = -1.118 \)
- \( \sec \theta = 1.357 \)

We need to find the **remaining ratio**:

### \( \csc \theta \) (Cosecant)
The cosecant is the reciprocal of the sine function. Since \( \sin \theta = -\frac{3}{2} \):

\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{2}} = -\frac{2}{3}
\]

Thus, 
\[
\csc \theta = -\frac{2}{3}.
\]

### To summarize, the trigonometric ratios are:
- \( \sin \theta = -\frac{3}{2} \)
- \( \cos \theta = 0.745 \)
- \( \tan \theta = -0.894 \)
- \( \cot \theta = -1.118 \)
- \( \sec \theta = 1.357 \)
- \( \csc \theta = -\frac{2}{3} \)

Do you want more details on any part of this solution, or have any questions?

Here are 5 related questions to explore:
1. How are the signs of trigonometric ratios determined based on the quadrant?
2. Why is the sine value \( -\frac{3}{2} \) unusual for real angles?
3. What is the relationship between the reciprocal trigonometric functions?
4. How can you determine the quadrant of an angle from the given trigonometric values?
5. How do you solve for an angle when given the values of sine, cosine, or tangent?

**Tip:** Reciprocal identities can be very useful for simplifying or solving trigonometric equations. Make sure you're familiar with how each function relates to its reciprocal.

Find the remaining trigonometric ratios of θ based on the given information. sin θ = -3/2 and θ is not in QIII. cos θ = 0.745, tan θ = -0.894, cot θ = -1.118, sec θ = 1.357, csc θ = ?

This problem demonstrates how to find all trigonometric ratios, including sine, cosine, tangent, cotangent, secant, and cosecant. It includes key concepts such as reciprocal identities and understanding the signs of trigonometric functions based on quadrants. Suitable for students in Grades 10-12.

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