# Relationships between Trigonometric Functions

• Angle (argument of a function): $$\alpha$$
Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha$$
1. Expressing the sine in terms of cosine
$$\sin \alpha = \pm \sqrt {1 – {{\cos }^2}\alpha }$$
Note: The sign in front of the radical on the right side depends on the quadrant in which the angle lies. The sign and value of a trigonometric function on the left side must coincide with the sign and value in the right side. This rule also applies to other formulas given below.
2. Expressing the sine in terms of tangent
$$\sin \alpha = \large\frac{{\tan \alpha }}{{ \pm \sqrt {1 + {{\tan }^2}\alpha } }}\normalsize$$
3. Expressing the sine in terms of cotangent
$$\sin \alpha = \large\frac{1}{{ \pm \sqrt {1 + {{\cot }^2}\alpha } }}\normalsize$$
4. Expressing the cosine in terms of sine
$$\cos \alpha = \pm \sqrt {1 – {{\sin }^2}\alpha }$$
5. Expressing the cosine in terms of tangent
$$\cos \alpha = \large\frac{1}{{ \pm \sqrt {1 + {{\tan }^2}\alpha } }}\normalsize$$
6. Expressing the cosine in terms of cotangent
$$\cos \alpha = \large\frac{{\cot \alpha }}{{ \pm \sqrt {1 + {{\cot }^2}\alpha } }}\normalsize$$
7. Expressing the tangent in terms of sine
$$\tan \alpha = \large\frac{{\sin \alpha }}{{ \pm \sqrt {1 – {{\sin }^2}\alpha } }}\normalsize$$
8. Expressing the tangent in terms of cosine
$$\tan \alpha = \large\frac{{ \pm \sqrt {1 – {{\cos }^2}\alpha } }}{{\cos \alpha }}\normalsize$$
9. Expressing the tangent in terms of cotangent
$$\tan \alpha = \large\frac{1}{{\cot \alpha }}\normalsize$$
10. Expressing the cotangent in terms of sine
$$\cot \alpha = \large\frac{{ \pm \sqrt {1 – {{\sin }^2}\alpha } }}{{\sin \alpha }}\normalsize$$
11. Expressing the cotangent in terms of cosine
$$\cot \alpha = \large\frac{{\cos \alpha }}{{ \pm \sqrt {1 – {{\cos }^2}\alpha } }}\normalsize$$
12. Expressing the cotangent in terms of tangent
$$\cot \alpha = \large\frac{1}{{\tan \alpha }}\normalsize$$